Supposedly, because just having the server up does not require any manual labor, but accepting a payments constitutes trade, which is forbidden during Sabbath.
(Nevermind the fact that an institution which entire purpose is to assist those in need is the most non-accessible organization ever - from not having accessible entrances to buildings, to their payments gateway which supports IE only. Yes, IE is the ONLY browser officially supported).
To accept a payment from a Jew on sabbath would definitely be wrong. So an Israeli site, which can expect that Jews will use it, must shut down on the Sabbath.
Leaving the site running, but not accepting money is probably the compromise they made because a lot of Israelis are secular.
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.