I think you might be able to store the session using
web("https://github.com/", JMP Window)
and logging in. Then you might be able to just use Open on the URL.
Open("https://community.jmp.com/kvoqx44227/attachments/kvoqx44227/sample-data/118/1/How%20Domestic%20Is%20My%20Car.jmp")
I think you can also do it interactively (not sure if this is recorded)
Other option is to look into github's documentation (maybe https://docs.github.com/en/apps) and utilize New HTTP Request as needed (I would go this route)
Ediit: Most likely better documentation GitHub REST API documentation - GitHub Docs
-Jarmo