Hopefully this is just a supply register and not supply return combo (LOL). A supply really needs some good grill with good air throw to heat or cool your house. This metal door will never work properly. The question is how much money you are willing to throw at it...

Sorry I do not understand floor furnace comment or the C shape either .Yes we need to know is this an actual heating vent or what ?

If it were me, I’d cut a piece of 1/2” thick oak slightly bigger than the hole in the floor, do a decorative bevel around the outside edges, notch it to fit around the register, stain it to match, and fix it in place. If this is beyond your skill level, get a carpenter to do it for you.

Thanks everyone! We found a custom floor register shop and are hoping they can cut one to size. @Patricia Colwell Consulting by C-shape I just meant the floor register has to go around the wall one because it sinks deeper into the floor. It would need to be a rectangle but with an area cut out to accommodate the wall vent. These are old vents, not used so we’re fine to cover them and hopefully seal in some of the warmth in our house.

Could you please share contact information for the custom floor register shop? Thanks