I have an old Schwinn Le Tour bike from around the 1975-1985 range. The rear derailleur broke and no longer holds any tension, so I need to find a replacement that will work. The current one is an "Altus" but obviously it is not made anymore given it is so old. A cheap one is fine as long as it will work for occasional use.. the bike gets 10 short rides a year. I don't have the bike with me now but it is either a 10 or 12 speed. Any suggestions? Thanks.
Pretty much any derailleur of similar vintage will work.
In the days of 5/6 speed rears, the chain was the same width so as long as the derailleur can move that far, it will work okay.
You could probably get away with a new low-end Shimano Tourney derailleur too, but they look a lot different and are clearly "wrong" for the aesthetic, if you care about that.
Given the bike is friction shifting, you won't have to worry about indexing the gears.
Your other option is to open and inspect the existing rear derailleur - I suspect one of the springs has failed by snapping or by loosing tension with age. It is possible a spring has slipped off a mounting peg too, that's not impossible.
You might be able to identify a suitable new replacement spring for a few dollars at the local hardware store - length and tension are the main factors, as long as it fits in the space available.
There are many roughly similar-looking 5/6 speed derailleurs with the "claw" type mounting (for a frame with no derailleur hanger), and they'll all work. It doesn't have to be old; they're still made today. Typically it will be a low-end or generic brand. Some may purport themselves to be 6/7 or indexing or SIS-compatible, etc. That's all fine if it's what you come up with. It just needs the claw. Even the current claw-mount Tourney would work, but it's bigger and floppier than you need, and availability may be scarce. Here are some that are still in production, but there are more. These are cheap, under $25US.
I would be looking at eBay to find old 5/6 speed derailleurs. As you probably don't have indexed shifting, the pull ratio doesn't matter. A 7-speed derailleur might work if you can't find a 5/6 speed derailleur.
The problem with modern derailleurs is that from 5/6 speed era, the cassette width was increased not once (to 7 speeds) but twice (to 8 speeds). The low limit screw that prevents the derailleur from jamming the chain into the spokes (generally ruining the rear wheel unless you use something called a "dork disc", which despite its dorky name is highly recommended), is probably not long enough to prevent chain-spoke interaction in 5/6 speed systems. If you replace the low limit screw with a longer one, it might miss its target because the target was designed to be at an optimal location generally where an 8-speed cassette ends, not where a 5/6 speed freewheel ends.