If it was a single web site, I'd suspect an expired security certificate on the server.
Otherwise I'd be checking for allowed protocols for the internet connection - I have no idea where to look for this on an Apple device.
For example I've recently had to help several people whose computers would not run a particular application - the problem was that the web based licensing system expected connection via TLS instead of SSL (after a security 'upgrade'), where TLS is not enabled on the computer no connection is made.
Sounds like one of those things that'll be simple when you find the answer, but may be a PITA to locate the cause.