How do you know it is a 1951 truck? From the appearance of the truck or from a registration or title or from what the owner/seller has told you?

You photo-link does not work.

50% of my four 1954 trucks had original titles/registrations using the vehicle ID and 50% have original titles/registrations using an engine ID (both ID locations were used in two states, VA and WA).

The main thing to be concerned with is what ID is on the title.

The next thing you have to be concerned with is if your state looks at the IDs on the truck when you register it, especially if your state looks at the cab ID, which you say is missing.

Nonetheless, if the title/registration matches the ID on the engine, you will probably be OK, no matter what state you are in. When you see what the owner has, you could then contact your DMV.