Can I travel without an Approval Letter and still get a visa when I arrive?

No, you cannot. You are not allowed to enter Vietnam without a visa approval letter unless you are in the Vietnam visa exemption list.  In order to board the plane, you have to present the approval letter to the airlines. This letter is also compulsory for getting visa stamp at Vietnam airport. Therefore, we recommend you to spend your valuable time to apply for visa on arrival to avoid delaying or changing your flight.