Yes, a dealer can have you sign a purchase order and submit it to your bank for approval before they sign and finalize the sale.
Once signed, purchase orders are legally binding and provide an itemized list of every cost and fee part of the transaction, so the dealer wants to make sure your loan has been fully approved, and everything is in order before signing.
The dealer was right not to sign, and you also shouldn’t agree to a purchase order until your bank reviews the agreement, fully approves your loan, and provides you with a check addressed to the dealer. If any changes are made to the purchase order, you should have your bank review the changes and confirm the following are correct: