To get reimbursed for a DITY (these days it's called a PPM - personally procured move) you will need to sit through some classes and sign a bunch of papers. Call them first.
You might be able to do the move first and then submit your receipts. If you do, there's some risk you'll screw up some detail. Be sure to get THREE vehicle weights at a certified scale: empty at origin, full at origin, and full at destination. This is a fairly new anti-fraud requirement. There is other fine print (eg a towed vehicle isn't part of your reimbursed weight).
The best thing is to call them and ask. You're certainly not the first person to be in this situation.
I do wonder though, if you're just coming into active duty if you can get reimbursed for a move you executed before you were actually on AD, even if that move was to the first duty station.