Yeah be patient. This coronavirus vaccine rollout is the first that isn't being coordinated by the government as we shift to normal commercial vaccines. CVS and Walgreens have had supply delays. Also as providers are now on the hook for the cost of unused vaccines, they've had to estimate vaccine demand. Likewise there have been insurance billing issues as insurers set up billing for the new coronavirus vaccines. Just remember that you shouldn't have to pay for these vaccines- insurance should cover them and if you are uninsured there is a government bridge program at participating locations. Just make sure to verify ahead of time.
These early issues are already starting to clear up, so it should be better in the weeks ahead.