Partly. BULKHEAD's FAQ says controller support is limited in the current version, and the Steam store lists no controller support. Play on mouse and keyboard if you can.
Controller buttons cannot be rebound in game, and keybinds may not save. On some PlayStation controllers the LB and RB yaw bumpers do not work. Fix that with DS4Windows: remap the bumpers there, outside the game.
Controller settings can be applied to vehicles only, so a common compromise is flying on a pad and shooting on the mouse. Default pad flight: right trigger collective up, left trigger collective down, LB and RB yaw, left stick pitch and roll.