Immediately upon updating to iOS 11, I had this same incredibly frustrating problem ("iPad can't connect to App Store"). After 20 hours of vain attempts to fix the problem, including 10 factory resets and conversations with 10 different Apple tech support people, someone at a local Apple Store finally figured out a solution:
1) Plug the iPad into a Mac.
2) From iTunes, do a backup, then do Restore (to factory settings).
3) After the iPad restarts, set it up as a NEW device. Do not restore from backup.
4) Now here is the most important part. When the iPad asks you to enter your apple ID, do NOT enter any ID. There is a fine print option at the bottom of the screen that will let you continue to the main screen without entering an Apple ID. If you enter an Apple ID when the iPad starts up, you will go right back to "iPad can't connect to App Store".
5) When you get to the main screen, open up the App Store. This time it should load properly.
6) Download any app you like. When the App Store asks you to login, now it's OK to use your Apple ID.
7) Restore from backup.
This bug in iOS 11 caused me so much hassle that I am not updating any other devices to iOS 11 -- once around the block was enough. I hope the solution above is helpful to other people.