Can you convert a 2 prong plug-in light into a hard wired fixture?

Sure, sounds like a creative way to use an old favorite table-top fixture and save a few bucks too. There would be a few things I would consider.

First and foremost, be sure the "2 prong plug-in light" is made for outdoor use if you plan to mount this outdoors!

Secondly, how are you going to securely mount this fixture to a ceiling or wall?

Designing a custom base, may end up being more trouble than it's worth. After installation the fixture should be flush against the surface, and everything tucked inside.

Next, I wouldn't forget, or leave out the ground. Even it is a 2 prong light. If your going to mount this on a wall, ceiling, or anywhere, chances are you already have a hole cut out and "2-conductor with ground" Romex type wiring ready at the instillation area. Where to attach the bare ground wire, maybe too risky if the fixture is not an metal chassis.

Note: Assuming that there will only be one fixture on this branch circuit, then go with a gauge of Romex similar to that of the fixture cord. If using many in series or parallel , reference a NEC handbook for detailed guidance.