First, I would create extremely fine grained but easy to check eligibility tiers. Going by date of birth would be the most basic solution. At any given time, vaccinations are open to people born on or before a given date. [0]
Then, I would set up real mass vaccination sites. None of this almost-all-vaccines-come-from-big-medical providers nonsense we have in most of California. The CA scheme is fundamentally inequitable -- I got an email from a medical system that told me that they would email me when it was my turn if I had seen a primary care provider in the last three years or had two specialist appointments in the last three years. This fundamentally discriminates against people without adequate insurance who haven't seen a doctor in a major medical system recently.
Mass vaccination sites would be open to anyone with a date of birth <= one threshold on a walk/drive-in basis with no appointment and with a different threshold for appointments. Optionally there would be another threshold for people who want to hang around and see if they can get an extra shot at the end of the day. I would adjust the thresholds as needed to keep occupancy at the level I want it. This requires minimal IT infrastructure.
I would let major clinics administer vaccines with whatever appointment system they want, with some rules. First, the clinics may only vaccinate people who have been eligible at the mass vaccination sites for some amount of time, say two weeks. So if you want your chance first, you go to a mass vaccination site -- clinics are just for convenience. Second, the clinics only get more vaccines when they are administering an adequate amount of the vaccines they have already received.
I would track people who have been vaccinated with a database with no integrity constraints whatsoever. Each site submits a trivial machine readable form with a list of people they have vaccinated. This list gets imported to a database. Dealing with all the inevitable issues is not permitted to block actually giving vaccines. [1]
Finally, everyone who gets their first dose gets a printed (or hand-written!) card with the date of their next vaccine and a reference number that can be used to look up the card in the database. This reference number is (date, site, line at site, sequence number) -- these are supposed to match up with the end-of-day logs, but they require no central coordination. [2]
If you have a card for a second shot, it is valid at any site once it's the right time.
Damnit, this whole scheme could be done entirely without computers, and that's how it should be. It doesn't even need two-way communication -- there could literally be a section of the newspaper saying "if you were born before m/d/y, come to Dodger Stadium or Levi Field or ... and get your vaccine, no appointment needed. If you were born before m2/d2/y2, you may call this number of visit this website to make an appointment." If the whole IT infrastructure goes down, then it will be annoying but everything will continue to work.
[0] There are many possible improvements, but they should be kept simple. For example, people born on or before a given date or people with a doctor's note explaining that they have some risk factor.
[1] Pay some contract development firm to develop a nice app that scans an ID card, does all the data entry, and prints out a receipt. The app has one primary requirement: it does not print a receipt until the server acknowledges receipt of the request and commits the transaction. But make sure that, even if the app fails completely, everyone at the clinic knows how to use a piece of paper and a pen to do the same job. Or a laptop running Excel or LibreOffice -- there is no shortage of laptops around. Or a Google Sheet, etc.
[2] Of course you can cheat, but most people won't. And the logs will mostly reveal the cheaters.