Not in California of course. However, we use our own fuel blend and are generally reliant on Central Valley oil sent by a different pipeline operator and Alaskan crude shipped by tanker.
The issue here isn't that they'll run out of fuel immediately. The way it works is that the local fuel "depot" is filled several ways, but by pipeline is the fastest, cheapest, and (usually) most reliable. They might still be able to ship by rail tankers, sea tankers, and trucks. But that's a lot more work. Preferably truckers are only doing short trips from these fuel depots to gas stations.
I believe that it hasn't stopped fuel depots from operating if they still have fuel. But this pipeline moves up to 2.5 million barrels of fuel per day and eventually they're going to run out. The biggest issue may possibly be with airports. A lot of airports are directly serviced by pipelines.