Where I used to work we had refuelling agreements with multiple vendors. Yes, hospitals and critical infrastructure are prioritized first, but we ran on generators for an extended spell during the Northeast Blackout of 2003[1] (the outage affecting our facility only lasted a couple of days, but due to the utilities' appeal to reduce load on the grid until it was fully functional we decided to run on generators for over a week).

Edit: One of my fond memories of that time was actually greeting the refuelling truck and directing it to the point where he could plug a hose into a pipe on our building and start pumping.

[1] https://en.wikipedia.org/wiki/Northeast_blackout_of_2003