Fron JKR interview
Q: Since Hagrid's name was cleared in Book 2, will he ever be allowed to do magic openly again ? (Jan Campbell)
A: He is allowed. He has been allowed to do magic openly ever since he became a teacher but because he was never fully trained his magic is never going to be what it should be. He is always going to be a bit inept. (src)
Basically, until PoA,Hagrid was prohibited from doing magic or having a wand (by Ministry decree, from what I can tell from many sources none of which seem canon). However, since JKR used the words "been allowed... since he became a teacher", it follows that he hasn't been allowed before then.
After his name was cleared in CoS and he became a teacher in PoA, he's allowed to have a wand. Why he doesn't bother is a different question :)