How long can most DCs run with just the fuel onhand?

There really is not a universal answer for this. Every generator will have what is called a "day tank" that as you might guess lasts for one day under a nominal load.

The day tanks are connected in pods to large diesel fuel tanks. Every {n} number of generators get a main tank. Those tanks vary in size depending on how much the company wishes to spend and how resilient they need to make their data-center excluding fuel trucks. Cities have regulations about how much fuel can be above or below ground at each location. My main tanks were 10K gallons. Each generator used over a gallon per minute under load.

And you are right, during a regional or global disaster fuel trucks will be limited. They who bribe the most get the last fuel but that too will run out. Ramping up production and distribution takes weeks and that assumes roads are still viable and the internet outside of the data-center is still functional.