IPhone with VPN

Question

Is there a way to set the mail app on the iphone to automatically use the VPN connection without having to manually turn on VPN everytime you need to check the mail?

Answer 1

Check with your school networking folks.
It's quite possible to substantially lock down a network at the firewall and at the switches used in larger networks, and to monitor what's not locked down for purposes of security, accountability and compliance.   (VPNs can be a means for circumventing established security, so it wouldn't surprise me to see those ports blocked from primary schools, and from various other security-conscious organizations.)
It's also possible that there's a misconfiguration within the network, and that's blocking access.  Multiple layers of NAT can play havoc with various VPN connections, for instance.  (IP networks can be large and complex, and errors happen.)
Ask.  That approach will also reduce the changes of getting in trouble with the school if the blocks are intentional, and can potentially also help the school networking folks sort out any unintentional network misconfigurations.  Why?  In general, intentionally circumventing network blocks can be viewed... dimly... by various organizations.  Sometimes just dimly, sometimes with administrative consequences, and sometimes with legal consequences.
But since you have LTE mode, you can use that, and avoid the school network entirely.
As for your question, the iOS clients provide L2TP, PPTP and Cisco support, and those protocols have fixed port assignments.  You'd likely need both a different client and a different VPN server, or command-line or port-forwarding connections with ssh on if-it-is-open TCP port 22 and related, but — if the port blocks are intentonal — that's just asking for administrative problems if (when?) the traffic gets noticed. 
Ask.  If you ask nicely and if the blocks are intentional, they might even open a route for you.  If the blocks are not intentional, you can help test the configuration corrections.