How is ammo limited, if at all?

There is no limit to ammo, although there is a limit to magazine size. The weapon indicator at the bottom right will show how much of the magazine you have used up so far. Note that there is not necessarily a whole number of shots per magazine. Laser Rifles, for example, will have a little bit of indicator left but still won't be able to fire.

Reloading ends your turn to reset the magazine to full. This is free other than that, and you can do this as many times in a battle as you like. You do not have to buy ammo, and a soldier will never "run out" of ammo during battle.

Rockets, on the other hand, are limited per battle. This only applies to normal weapons.