First a note on the naming. Unfortunately, many tricks have fallen pray inconsistent naming, and this is one of them. Generally, when you heard or read "Vertical Loops", what is meant is "Vertical Punches". Many people assume that vertical loops would refer to inside loops done upward, and I think rightfully so. I myself try to refer to the two kinds as "Vertical Inside Loops" and "Vertical Punches" or "Vertical Punching Bags" to try to eliminate some of the confusion. I urge you to do the same thing when you can so that people know what's being talked about.
You'd want the same kind of yo-yo that you are using for inside loops or regular punches. Generally a non-butterfly yo-yo with a fair amount of weight in the center.
As with the other loops, it IS possible to do vertical punches without the yo-yo flipping each time, and in those cases a butterfly would be easier to control, but I do not really recommend going that route myself. It has been my experience that non-flipping loops tend to be more prone to going suddenly out of control, and harder to modify (like tilting on purpose). So, if you can make a dual-colored yo-yo to check the flipping, it is recommended.
Something different from regular punches is that you may want a yo-yo that loops slightly upward. In regular punches, the yo-yo is always trying to fall, and so a yo-yo that loops up (with inside loops), will tend to fall during punches and be very hard to keep up. But in vertical punches, the yo-yo is going to be falling down toward your hand in any case, and a yo-yo that loops up may give you more control.
But as with anything, don't overdo it. A yo-yo that loops really high will be difficult to control and feel more sluggish and straining. My main reason for mentioning it is that a yo-yo that loops way down will be very hard to control in vertical punches unless you can go very fast. If you notice your vertical punches being easier after a bit of practising, and then harder to do again before they break, it is probably from the thinning of the strings making them loop higher and higher. An adjustable gap yo-yo may help you get a feel for where you're most comfortable.
As with hop the fence and regular punches, you want the angle on the right hand to be about 1 o'clock and the left hand to be about 11 o'clock. But this time it is as if the clock was on the ceiling. It should be possible to bring a yo-yo up from hop the fence to regular punches to vertical punches (called the Fountain transition), with the yo-yo keeping approximately the same tilt from your field of vision.
One of the most common problems with vertical punches is that of the tilt falling over to the side with every punch until it goes out of control. Some things to check is that you're wrist motion is strong enough, that the yo-yo does not loop too far down during regular loops. You might try doing your wrist motion sooner (before it reaches your hand) or going faster in general.
Also, remember that you probably had a similar trouble with inside loops, and it may take a lot of practice to overcome. You might also try compensating slightly by making your wrist motion more inward instead of straight forward (more to the left for your right hand and vice versa), with the angle of the yo-yo more toward 12 o'clock.
You may have more strength in that position and might help the yo-yo from tilting out of control, but if you find that the yo-yo is still slowly tilting outward, then it might be that something else is going wrong, and all you're doing is buying a little more time without fixing the real problem. Also, if you tilt inward too much, you might find not be able to do it two-handed (usually called the Bicycle, since it looks like a kid laying on their back moving your legs like riding a bicycle) without hitting yourself.
That's all well and good, but you might be wondering what the best way to start them is. There are various ways to transition into them from other tricks (a warm start), but here I'll describe the two main ways to get into them from the yo-yo in your hand (a cold start).
I'll call the first method the regular start, as this is generally the main way that they have been done, and the way most people still do them. You throw a high forward pass, and have it swing under your hand. This is essentially the first motion of a reach for the moon (does not flip), but you'll want to make it go a bit more forward than usual, so that you can go into the circular motion of the vertical punches.
The main thing to keep in mind here is to keep an eye on the tilt of the yo-yo. When you do the first motion, you want it between 12 o'clock and 1 o'clock (for the right hand). If you keep it perfectly straight as in a moon, it may take a couple of punches for the tilt to even itself out. If you can tilt it a little bit during that first motion, it'll make it a bit easier to get into the punches, but be really careful not to overdue it. Remember that one of the main problems is the tilt falling on vertical punches, so if you start off with a 2 o'clock angle, it's going to be really hard to get control of it again.
Also, keep in mind that the first motion is a set up more for the vertical punches and is a different from the regular motion. Going from the non-flipping moon type motion into the flipping punch motion may take some work, and the dual colored yo-yo is again recommended.
If you read the regular punching guide, you may remember that I didn't recommend throwing out a forward pass and going into them from there. Instead I recommended going up from the hop the fence. The reason for the difference is that the pull of gravity makes a non-flipping transition fairly hard to do straight in front of you, and you're likely to get messed up in the first couple rougher punches transitioning from the non-flipping to the flipping motions.
You can do a much more relaxed transition above your head. Remember how slow reach for the moon can be done. Also, transitioning form hop the fence all the way up to vertical punches is much harder than just up to regular punches. The hardest part of the fountain transition is generally the high point right before reaching vertical. However, the fountain is still a good trick to practice, and if all the other methods fail, you may be able to get a feel for vertical punches by that method.
The second main way to get directly into vertical punches I'll call the quick start, as it is much faster. This has come into favor lately by a lot of the more skilled players. It is more difficult than the normal start, but works very well.
Basically what you want to do is throw a throwdown straight up (no pun intended). You want to curl your arm so that your elbow is pointing up, and your hand is near to your ear (incidentally, a good position for throwing a very fast sleeper). Basically you'll keep your upper arm where it is, and use your elbow and wrist to propel the yo-yo up and a little forward. From there you can go directly into the vertical punches.
Make sure you're throwing hard enough that the string fully extends. Also, you may want to throw with a slight tilt to help you get into the punches. If it always seems to go too far forward, try throwing harder or throwing a bit behind you, to get it to go up to the right spot.
One thing to keep in mind is that while vertical punches still need a lot of force to keep going, generally it will be more wrist motion and less major arm movements. When doing regular punches, there is always that great tendency for yo-yo to fall, so you must use a fair amount of arm to follow it downward each time and then provide the necessary lift.
In vertical punches, the yo-yo will be falling back down toward you, so you don't need nearly as much full arm movement. Something to keep in mind if you are getting to vertical punches from a Fountain transition, or if you're just really used to doing regular punches. If you find that on vertical punches, the yo-yo keeps going farther and farther behind you until you can't control it anymore, you're probably using too much arm and lift motions.
You don't have to keep your arms perfectly still, or use entirely wrist, but I find that any arm movement, while forceful, don't actually need to move that far to be effective. It is more to give the motions some extra "oomph".
While hop the fence, punching bag, and vertical punches all are very similar to each other, they do use slightly different arm and wrist motions, so you need to be able to "switch modes" when needed.
Keep in mind that the yo-yo is angled up enough that your palm will be up when you catch the yo-yo. But this is different from the standard palm up catch that you'd be using for a forward pass, and in fact you may never have caught a yo-yo before the way that you need to for vertical punches.
Your arm pointing up, and your wrist is bend back a bit so that your fingers are pointing a bit toward you. Will probably feel a bit strange at first, but catching it this way is better so that the yo-yo will be the right way in your hand to throw.
Also, you may need to fight the urge at first to send the yo-yo back out like the second half of a reach for the moon, and catch it that way. While there's nothing wrong with that, it is good to be able to do it either way.
This is partly a personal preference, and also the set up on the yo-yo. If the yo-yo loops up a lot (during regular looping), it'll tend to go forward a bit and it may be easier to control if you lean back further so that it is traveling straight up.
While having anything flying above your head has some danger, as long as you're careful, you should be ok. The main thing is to try not to use a yo-yo that's going to stall out on you at the end of the string, and not to bring your hand too close to your face. If you bring it too close, there's a good chance you'll get yourself in the chin as it comes around.