I learned a lot on the Minneapolis Vaccine Hunters Facebook site. I created an account just to access this, got the info I needed, and killed the account.
Some details that probably apply nationwide: CVS opens appointments around midnight, Walgreens about 6AM. Independent pharmacies, Walmart, and many grocery store pharmacies also have the vaccines. I was able to get an appointment at a small pharmacy about 2 hours away, but if I'd had more time to work on finding a local appointment, I could have.
The CDC has a "find my vaccine" link on their website which will tell you which local pharmacies have the vaccines (and which vaccines).