They shouldn't need to. The issue is them accepting your order during their observance of the Sabbath. If you're in a different timezone and are still supposed to be observing the Sabbath (while the website's owners are no longer in their Sabbath window), they'll accept your order; you've violated the Sabbath, not them.
(At least that'd be my interpretation based on my often-limited understanding; I'm not Jewish, so I'm hardly an authority.)
I'm not sure if their automatic system accepting the order is a problem. It shouldn't be (no people are involved in any of it from their side). However they serving someone who is violating Shabbos at the same time is a problem (for both them enabling the violation, and the violator). And that's timezone dependent.
It is dependent, but it's easier to prevent all transactions in such case than trying to ask the user about it and rely on honest answer. Plus it would probably make it "moris ayin", i.e. the likelihood of violation to the observer which is a whole long subject on its own. I'm not an expert in the law however, so those who implement such systems would have to ask about it anyway.
(At least that'd be my interpretation based on my often-limited understanding; I'm not Jewish, so I'm hardly an authority.)