Is there oil in it now? Most of those types of transmissions either use regular old gear oil or grease. If it's straight cut gears and bushings it'll work with just about anything for lube, look at the seals if they are lip seals and roller bearings I'd be inclined to fill it up with 80w-90 and see what happens If there aren't any seals I'd be looking at corn head grease. I would think the worst that could happen if you fill it with oil is that it would leak, look for something in thats rated for manual transmissions and good for bronze synchros which won't hurt the bushings. GL-4 rated 80w-90 would be my best guess.
Those old tecumseh motors were used on a lot of things and they are pretty bulletproof, I have an ice auger with one that is at least 30 years old and it's dead reliable all the way to -40F. Loud but it runs and will probably still be running when the world runs out of oil. They rated them around 3-3.5hp depending on who's rating you look at and I think somewhere around 85cc.