There are a number of brackets that hold the vent window in place, one of them at the sill of the window and another at the bottom of the window track on the inside of the door, as well as a couple tabs for screws into the front of the window opening. The inside trim is also notched where the division bar on the back of the vent window goes through it. You'd also need to replace the window regulator, as they're a different style between 50 and 51+. There's also a rain diverter that covers the front edge at the top to keep rain out if the vent window is slightly opened.
If you have your old '51 door you can probably salvage the brackets and window regulator from that. But it might be easier to restore/repair your original door.
Here's a couple pix of the vent window assembly, and the bottom of the window opening showing the bracket at the window sill.
Kevin 1951 Chevy 3100 work truck Follow this saga in Project Journal Photos 1929 Ford pickup restored from the ground up. | 1929 Ford Special Coupe (First car) Busting rust since the mid-60's If you're smart enough to take it apart, you darn well better be smart enough to put it back together.