Is it ok to use LED headlights with a dust cap?

Covering up bulbs for what reason? If the bulbs came with covers it probably was just to protect them during shipping. Post a photo if you want a better opinion.

On the back of the headlight housing assembly, there are caps that protect the inside of the assembly from dust and moisture. However, this means there is not a constant supply of fresh air to the bulbs, and so it seems possible they would overheat

LED's should use less power than the standard bulbs which means less heat. You need those caps to keep moisture out.