Can a repossession agent take a vehicle without waiting for the borrower to show proof that payments have been made?

Yes. It is perfectly legal for a repossession agent to take possession of a vehicle when they are acting on behalf of the lender. The repossession agency does not have the option of allowing the borrower to retain the vehicle even though proof is presented that payments have been rendered. Such issues are strictly between the borrower and the lender. The lender and/or court being the only parties that can rescind the repossession action.