Can I get a Non-O business visa for running a company in Thailand, and what documents are required?

The business visa is generally a Non B visa. You can get a work permit on a non O if married to a Thai national or have Thai kids but I’m assuming you mean the Non B.

For most companies trying to get a Non B in country is nigh on impossible. Many can’t provide the vast amount of paperwork immigration would need to issue one. And I mean vast.

Usually it’s way easier to have the person visit a nearby consulate and apply there (or via the evisa system where applicable - you have to be in the country of the consulate you are applying to regardless that you are using the evisa system).