ex-refinery guy here. The picture is actually very complex, many interdependencies. Each refinery is pretty much set up to run optimally on one type or mix (which may or may not also mean one or one set of sources) of crude oil. The flexibility of a given refinery to do something else varies with its particular equipment but profitability (ie prices go up) will almost always suffer, at least short term, if they are forced to switch crude slates. In the US, most midwestern refineries run Canadian crude or oil that comes by pipeline from the gulf. East coast refineries import more, West cost refineries source some local (CA is a big producer but nowhere near CA's consumption), a lot of Alaskan, some ME. Refiner's generally hike their prices the instant world oil prices go up, though in reality they have several days (not months) supply already on hand and any oil at sea may already be paid for, though that varies also, some tankers do arrive with the oil un-purchased and the deal is cut on arrival.
I'm few years out of the biz now so I don't have any inside sources any more, but I would estimate that in the US, total imports from non North American sources are low enough that given the fall in demand that the increase in prices will produce, I doubt we will see outright US shortages, but don't ask me to bet on that, and there will likely be some local dislocations.
The situation in Europe and in particular Asia is much different. They know their vulnerability to supply upsets so in general Asian refiners keep a LOT more crude inventory (month+) on hand, which is the only reason why you haven't had Asian nations (i.e. China) making more noise about retaliation against the US (trade etc) if this doesn't end soon.