I do not know about the 37 Ford glove box, but the 36 Ford glove box door can be installed fairly easily.
Place the door and the hinge plate inside the box. Install the bottom screws with acorn nuts (note previous photo showed screw ends inside the box, which I believe is not correct) inside the box loose as possible.
Then with someone holding the glove box door, install the springs on each end of the door with the door closed as much as possible. Then tighten the screws into the acorn nuts and the door should then be able to be closed and stay closed.
Hope it works for you like it did for me.