The other reason I just thought of would be that you lacked TUs to reload, although I doubt that's it. What happens with your reload failure? Does the spare ammo indicator show a non zero number (if zero, the ammo isn't recognized). When I try to reload but lack TUs to do so I think there's an error sound.
Edit: How do you reload? I typically click the spare ammo icon, but there should be a hotkey as well.
For a crummy work around, (ab)use one of the rocket launcher bugs: Revert to a save before the mission, bring a rocket launcher (or several) along. If they're too heavy, drop them in the drop ship and then return to pick them up when you need to dispose of the tank. If you put the rocket launcher in the inventory after firing, it gets reloaded automatically, so you can fire it again the next turn.
A less cheaty version would be to bring extra weapons of a type you can reload along (or extra lasers, but I doubt you've got spares), and use those to *slowly* whittle away on the tank (you need lots of ammo for ballistics, although a Hunter with a Division machine gun can probably destroy a tank eventually, if you can keep if from being destroyed).
The trick to bring spare equipment and dump it on arrival is rather handy, by the way. It costs TUs on the first turn, of course, but I'm going to use it to bring spare shields.