You dont need internet at a shul to have e-mail. They could have e-mail and access it elsewhere, especially if its a Yahoo/Gmail/Hotmail type account as most of them are
While you might see
Miryeshiva @yahoo.com
you likely wont see
Info@ miryeshiva.edu (or .com)
(I do not know if any of these are real or not, they are for example only)
And even if you see miryeshiva. edu meaning they own the domain name, they might have a mail server set up to get the mail remotely