My wife and kids were included on my orders to AIT, PCS move. Am I eligible to request my command for leave to move them?

Part of your in-processing will be travel and housing, which will help you sort that out. Depending on your class start, you wont be able to take leave to assist in the move, but the army/transportation will pay a moving company to pack, move, and unpack for you. It's actually a real relief.

Normally the Army allows you time at the losing and gaining station to conduct business in regards to PSC move. Travel time on the road is a freebie. I think around 350 miles in a POV constitutes one day of travel time. Try to get Permissive TDY before you leave. It will give you 10 days of lodging and meals for your family in regards to PCS moves. It can be broken up by lets say 5 to 5 days or 3 to 7 days from the losing to gaining station. You can ask for Permissive TDY at the gaining station but its not a guarantee. I also recommend that you contact the transportation office to arrange for shipment of household goods ASAP. Will you have a Government Travel Card? Try to get that card. You can do a partial Do It Yourself (DITY) move by coordinating with transportation. Essentially you weigh your vehicle full and empty to determine how many pounds you are moving yourself. You will get paid for moving part of it yourself. If you have more questions we are here and you know where to find us. Good luck. One more thing. Move all your extreme valuable items. Video tape what you own for loss and or damages.

Permissive TDY, perhaps. Also if you have a leave balance, you can request that too.
Just be aware that neither are an entitlement, and they can say no.