There should be nothing special about a VPN -- at the same time, I have no idea if your VPN does the right thing.
AquaMail doesn't care, really. It just uses standard networking functions to do what it needs to do.
My WiFi router is behind a VPN essentially -- L2TP, that's how providers set things up where I live -- no issues.
A VPN on the phone should be directing network traffic in a way that does not require all applications to be modified.
Things I'd check right away -- is whether mail ports are forwarded (and not just web browsing ports).
IMAP: 143, 993. POP3: 110, 995. SMTP: 25 (rare), 587, 465.
If you have any @gmail accounts in AquaMail with OAUTH, the app will also need access to:
https://accounts.google.com and
https://www.googleapis.com if it's an account *not* present in system settings
and
whatever Google Play uses to do its OAUTH thing if it *is* present in system settings (I have no idea what "endpoints" Play talks to).