If your car has been repossessed and you offer to pay the amount due in full, can the lender force you to reapply for the loan?

If you're paying the full balance, there would be no need for a new loan; but if you are only paying what's past due, the lender can probably require some additional assurances before releasing the vehicle back to you.

If you are talking about only paying the amount that was overdue/late (e.g. the 1 or more missed months behind) AS OPPOSED TO the entire balance that is still due, no it is not a given that the lender has to accept your payment that and continue with the loan like normal. It would depend on the terms of your loan contract, but it is very common that once in default the Lender can accelerate" the loan (require the entire remaining balance due). It is reasonable that the Lender may want to now reassess your financial situation, through a loan modification or outright new loan, since income issues may be why the loan fell behind. But you should definitely have an attorney look at any new paperwork you are given before signing just in case

No they can't.