I have this metal greenhouse frame that came with my house, and I was thinking of fixing it up. Any tips/recommendations?

the gold standard for plastic on frame is 6 "mil" aka 6 mm thick clear polyethylene film "greenhouse plastic sheeting" for something that small, I would look at a series of removeable clips to fasten it and unfasten/remove at the end of season. plastic gets holes and eventually degrades in the sun after a number of years.

for fastening, in the states we have a product lovingly referred to as "wiggle wire" that uses a precisely pre-bent stiff wire along a metal tracking to hold plastic into place. I think they make smaller gauges for smaller houses like that, because the stuff I have used would be overkill. and really, your project is small enough that like a box of those heavy duty binder clips from an office supply store would probably work. I would make the back+sides more or less "permanent" and make the front be made to easily open/seal for access, watering, and venting on bright sunny warmish days.

edit: this is what I am picturing when I say binder clip, but really once you get the material and get a feel for its durability, you will know what will work. 6mil poly is no joke. like I wouldn't puncture it, but you can clamp the shit out of it.

edit2: the other thing I would recommend down the line once it's rigged up, would be to get your hands on some "injection molded" trays (as opposed to the flimsier, "blow molded" trays). you wash and reuse them and they would make life easier with that existing shelving in there by not drooping or breaking/falling through. I like to think I'm a gentle, deliberate person, but early morning watering can get clumsy).

heavy-duty zip ties are a cheap way to reinforce the pins holding it together if it feels shaky. also, depending on your climate, having a plastic cover you can remove or peel back during high temps is easier on some plants than one permanently affixed to the frame (plus, as others have mentioned, the plastic will degrade due to exposure over time).

also, i usually place something with some heft (larger potted plants, unopened bags of soil or fertilizer, or bricks, etc.) at the bottom to help stabilize the set-up from wind, pets, and such. have fun!

Polytunnels seem common enough to legitimise plastic as a covering material. And, if it's thick enough to withstand the weather, then the plastic could be wrapped around the frame and riveted back to itself. That'd avoid problems with glue/clamping/etc.

You should be able to find a replacement cover, something similar to this, that you can just slap back over it. Be a lot easier and probably cheaper unless you already have the plastic lying around.

I'm so tired I read this as mental greenhouse and was like yeah let's fixit