Most likely not. He will not have leave between BootCamp and A school so if his command allows him to have it someone else will have to take it to him. It is completely at the discretion of his command and he WILL have to earn the right to drive. Seabee A school is not that long and then if he's going to combat training that takes place in MS. He will not know until near the end of A school where he's going to be stationed and if they will allow his car as well.
My Bee is overseas and it would not have made sense to have a car before he got overseas. He's just buying one there. Much less hassle.
I know you are looking for a clear Yes or No but the correct answer is: It depends. Better to assume no car than to count on having one, but be ready to take it to him if they say yes.