Lots of people write apps that connect to their web servers. I think it's pretty unusual when an app doesn't do that.
Internally, Apple is really old-school, big-iron corporate. I don't know specific details of specific products, but Apple's a big company, so there are lots of them. Generally speaking, if a 3rd party corporate networking/security app was a Big Deal 15 years ago, Apple probably still uses it today.
Probably the two most important things to check are IPv6 and SSL. Make sure your app works in a purely IPv6 environment. Curiously, this policy is 9 years old now. Apple has a solid track record of announcing major changes 9 years before making a hard change. People who ignore those warnings because their app is working fine then get blindsided years later.
Also, Apple can be very strict about SSL certificates for its own use. It really doesn't matter if your website works great in Chrome. It might not work from an Apple app or from an Apple internal network. People who roll their own SSL certificates are more likely to encounter this problem. I recommend checking your site with SSL Labs (http://www.ssllabs.com/ssltest/). Even if you get an A+, double-check the details against Apple's published specs like this one and this one.
I'm not saying these are the same requirements as Apple internal networks. Apple is a black box. I have no idea what goes on there. But this is something you can check without going through App Review. Send yourself an e-mail containing HTML with images hosted on your site. If Apple Mail won't display the images, then this could be the reason.
Topic:
App Store Distribution & Marketing
SubTopic:
App Review