Does anyone recognize this rock tumbler's make/manufacturer?

You don't need Mfg. to get a belt. It looks as though the belt pulleys are sized for a 1/2" wide V-belt [across the back or widest part of the belt]. Measure overall length over the top of both pulleys all the way around. Go to an auto parts or good hardware store and get a 4L pulley that has three numbers corresponding to the length you measured over the pulleys, i.e. if your measurement is 21 inches, the belt number would be 4L210. 4L belts are 1/2" wide, considered low horsepower belts, and dimensioned so as to be measured over the pulley tops. If your pulleys appear too narrow for a 1/2 wide belt, there are 3L [3/8" wide] belts available. If your belt supplier doesn't have the L designation belts they can crossover to a heavier duty belt numbering system for you. Note that you have length adjustment on the motor, start with the motor close to the driven pulley as shown in your picture for your measurement.

Rick take the old belt with you, the store (if it's any good) will have a gadget that can measure the old belt. It's probably stretched slightly so get one size smaller than measured.

Edit. I'm a bit concerned about that shaft rubbing metal to metal on the back of the disk, maybe it should have a bit of rubber on it.

Agree with what everyone else has said. You don't need to know the manufacturer to replace a belt.  Belts are generic, used for so many different mechanical things. Just need to get the correct size.

But still nice to know what you have. jamesp posted this photo on December 30, 2017 from an old advertisement. Pretty sure this is it. And AzRockGeek had mentioned the name SLIK.

Note the first feature mentioned is that it is an open type tumbler(gas will not accumulate) rockpickerforever.
The gas issue encountered long ago. Nothing has changed lol.
It would be interesting to see a comparison in performance of a slant verses cylinder design.
Viking's slant tumbler utilized a complex slant barrel that seemed to push the rocks thru a long rolling path.
Denise uses the Viking slant to tumble metals since metals create lots of gas.

Perhaps a short piece of rubber tubing pressed onto that shaft pauls ?

Is the machine small enough to carry in when you get your new belt. A really tight fitting piece of hydraulic hose would be my choice if the right size is available. A good parts house or really good hardware store could help with that.

I’ve used electrical heat shrink as a shaft covering with good results.