Finding a vein in a kid's arm works the same as grownups, but the tiny veins buried in the pudge can be slightly tricky.
First step is to get over the kid drama- it's just another vein. You want to get in there as badly as you want to get in any other vein, and they behave just the same.
Get your partner to hold the whole arm still, and look either in the back of the hand or the AC at first. If you flex the wrist so the hand makes a 90 degree angle at the end of the arm, there is usually a vein right in the middle of the wrist. This is the easiest one to hit on infants and neonates, IMO- and just like sometimes the veins under a person's watch arm more exposed than the open skin, the veins sometimes show better in the crease of the wrist of rubber band wrist babies.
In the AC, sometimes hyperextending the arm will help you see the vein and feel where it runs, but letting it flex a little when you stick makes it less flat. Make sure you have an armboard ready that's appropriately sized. Ask me how I know that trauma sheers will cut through a 3-ring binder in a pinch to make a tiny little armboard.
If no vessels are showing up at all, you might try a hot pack on those areas, or look in the foot. Again, they are all just veins. Don't be scared of them just because the patient is screaming at you. Stay cool, keep talking the whole time, and work quickly. When you find a vein, make a decision quickly about whether you are going to stick it or not, and then get 'er done.