Yes, NY makes it tough BUT, after jumping through some hoops, so to speak, I think you'll still be OK.
In the instructions for DTF-804, you see that they only have reciprocal agreements with eight states: Arizona, California, Florida, Hawaii Massachusetts, Michigan, South Carolina, & Wyoming.
But, if you'll look where it says "Note:" (about 1/2 way down, left column on page 2, instructions, of the form) it says the following:
Before completing this form, refer to the charts below to determine if New York State provides a reciprocal credit for sales tax paid to the state where you made your purchase. For motor vehicles, New York State provides a reciprocal credit for tax paid to a very limited number of states. If no reciprocal credit is available, use Form DTF-802, Statement of Transaction - Sale or Gift of Motor Vehicle, Trailer, All-Terrain Vehicle (ATV), Vessel (Boat), or Snowmobile, instead of this form. If you paid tax to a state for which no New York State reciprocal credit is allowed you should apply to that state for a refund
Then it says -
If the state (NJ) denies the refund you may be eligible for a credit for some or all of the New York tax you must pay now. To apply, file Form AU‑11, Application for Credit or Refund of Sales or Use Tax, and attach proof of tax payment to and proof of the refund denial by the other state.
Here's NJ's request for refund:
The form can be used for multiple types of taxes.
For refund of sales and use tax send to:
NJ Division of Taxation
Sales Tax Refund Section
PO Box 289