The CDC developed a testing protocol (crudely, a set of 3 conserved RNA sequences of the virus to chemically "grep" for). The protocol is here [1] and I think anyone with suitable equipment (an RT-PCR machine and an oligonucleotide synthesizer to make the primers and probes?) could run tests using just the information on that web page, though they would probably also want known positive samples of the virus to validate against. And there are lots of labs with suitable equipment. But as that page says, it is not legal to do this clinically.
The CDC also sent out test kits [2]. These are just vials of primer/probe pairs synthesized from the short sequences on the above page, and a fake positive control to validate with. These were authorized by the FDA, and so are legal to use. But apparently somehow they screwed up the production of primer #3 (or maybe the sequence itself is wrong), and so almost all the labs were not able to validate the test. After several weeks the CDC has still not fixed this problem, but labs are now allowed to run tests using just primer 1 and 2, so testing capacity is rapidly increasing (though of course who knows how sensitive or specific the tests are).
The federal organization preventing labs from running tests is the FDA [3], not the CDC. It is also the FDA who would have to approve the commercially produced test kits.
[1] https://www.cdc.gov/coronavirus/2019-ncov/lab/rt-pcr-panel-p... [2] https://www.fda.gov/media/134922/download [3] https://www.fda.gov/emergency-preparedness-and-response/mcm-...