No, they are not required to shut down their site by the letter of the law. However accepting money on the sabbath is not in keeping with the spirit of the sabbath, so I fully understand why they shut down.
A human is not allowed to work on the sabbath, but your machines can, as long as they run automatically.
Interestingly work animals are also required to be given the day off.
The main thing that is not in the spirit of the sabbath is accepting payments, even if by the letter of the law it's allowed.
For example, even though you could leave a physical shop open and unstaffed, and let people (non-jews) buy using the honor system, it's not the right thing to do.
However, leaving the shop open, so people can take things without money, and then they settle later is OK.
A human is not allowed to work on the sabbath, but your machines can, as long as they run automatically.
Interestingly work animals are also required to be given the day off.
The main thing that is not in the spirit of the sabbath is accepting payments, even if by the letter of the law it's allowed.
For example, even though you could leave a physical shop open and unstaffed, and let people (non-jews) buy using the honor system, it's not the right thing to do.
However, leaving the shop open, so people can take things without money, and then they settle later is OK.