Are we able to generate muons outside of a particle accelerator, or does all muography rely on cosmic rays?

It's pretty much just cosmic rays. I suppose you can sort of create them by using an accelerator to generate a beam of the appropriate particles that'll hit a target or decay and become a beam of muons outside the accelerator but that's not really all that practical. Incidentally, this is how neutrino beams are generated.

https://en.wikipedia.org/wiki/Muon_tomography:

“Muon tomography or muography is a technique that uses cosmic ray muons to generate two or three-dimensional images of volumes using information contained in the Coulomb scattering of the muons.

[…]

Since 2010s researchers are also exploring and attempting to use artificially generated muons—created by conventional accelerators or laser-plasma systems—for muon tomography.”

I may overlook something, but skimming the references, I get the impression the latter still is an idea. References are about simulating the machinery, discussing requirements of hypothetical machines, etc.

Muons are not stable, thus you cannot tear them off matter as you'd do with electrons. And they have a mass of 105 MeV each, which means you need a nice particle accelerator to create a few of them.

Furthermore, if you want (most of) them to fly in a particular direction, you need to scale that accelerator up.