I'd be inclined to say your pick is your pick, and that is your assigned vehicle which you are also responsible for.
However your commander can assign you any damn thing he please (it's up to the Munitorium to follow through of course.) And if you're on base then I would think it'd be pretty easy to check an alternate ride out of the vehicle pool. Obviously you're a rated driver.
The specifics are going to vary wildly by regiment and location of course. In some places they might let any one who wants one have a Leman russ, in others you have to fight a 5 round death match to get a skate board. One thing the IG is not, is consistant. Except for the suckage. That is consistant.