The vaccines are Pfizer (2 doses/shots) or Moderna (2 doses/shots). (They are not giving the Johnson+Johnson vaccine right now because they are doing extra safety testing.)
When you get your shot you need a photo ID. You don’t need anything else, but they might ask for health insurance information if you have it. If you don’t have it, it’s ok!
Make sure you keep your vaccination card. You can take a photo of it and email it to yourself as a backup.
It is safe for all immigrants to get the vaccine. No one will ask you about your immigration status, and you do not need to show documents about your status.