Wow, that's an old one. I thought it was when I saw the black sides - more recent Stingray's have been blue.
Nikon hasn't sold camcorders for at least a decade - maybe since the mid-80's. Since they were physically larger your camera should fit inside the housing.
But you're going to have 3 potential problems:
The biggest problem is going to be that the port on that housing is in the wrong place. The old Nikon cameras - like 8MM Sony's (OEM for Nikon at the time) - have the lens near the lower right corner of the camera body. Your UX7 like all the HDR series has it top center IIRC.
You have to be able to center your camera in the port otherwise you will get vignetting and other visual aberrations in your video. And you'll likely have to re-work the mounting plate somehow.
There's a slight chance that the port is high enough and the camera mass below the centerpoint of the lens on your camera is short enough, it might work with the camera offset in the housing. But I kind of doubt it.
According to Sony's e-support site, your camera has a LANC jack. If so, you'll be able to plug the camera into it for camera control. LANC was the only standard when that housing was made so there will be a short cable mini-plug inside the housing.
The issue might be that the electronics are pretty old so it's possible that even if it works, it won't be able to control all the camera functions through the jack - LANC has changed over the years. The old L&M housings have an exposed circuit board so if it was ever flooded, it will need to be replaced/updated. I think L&M still does upgrades.
The third problem is that your UX7 is physically shorter in length than the Nikon. So that will move your viewfinder farther forward in the housing. You can't open/see the LCD when it's in the housing.
Most of the older Stingray's had a semi-oval window on the top 3rd of the back plate so you should still be able to see something. It's also possible that if you found an old back-mounted Stingray LCD monitor, it could be replaced - it just latches on. I've seen a couple sold separately on eBay the past few years.
What did the seller say about the condition of the light batteries?
I don't think you'll find anything on the new L&M website about compatibility. Since they went to the new Stingray, I think the info was removed to be less confusing. IIRC correctly their manuals back then were pretty generic so I don't think they list model compatibility either. There used to be a really handy chart on the L&M website but I haven't seen it recently.
I almost did the same thing a few years ago - found a good deal on a Stingray on ebay also. But then I bought new instead. So some of the compatibility stuff is still reasonably fresh in my mind. I made an Excel spreadsheet listing all the models that were compatible - I may still have it archived - if I do I'll contact you.