If you do nothing, NY will find you within a few months after registration. When the letter comes, they begin with an inquiry of how much you have in the airplane and whether any tax has been paid. They will have looked for previous filings for sales tax and will have checked your personal tax returns to determine if you paid sales tax there.
You would file for ST-141 (
http://www.tax.ny.gov/pdf/current_forms/st/st141.pdf - instructions at
http://www.tax.ny.gov/pdf/current_forms/st/st141i.pdf )
You can pay as you go - keep copies of the filed forms and cancelled checks, etc. as well as the invoices that comprise the total for the aircraft.
I've seen people advise that you can avoid the tax by registering the aircraft as a XYZ-1 (rather than a RV-7) and claim that you paid the tax as you went along. However, they will typically respond by asking for copies of the receipts and want you to show that the receipts, collectively, comprise an airplane (aircraft, engine, prop, etc).
Dan