Do any of the M340 Ethernet modules have Email functionality?

Yes,
The M340 PLCs BMXP342020 with SV>V2.0 & BMXP342030/20302 can be configured to send an email out of the embedded Ethernet port on the CPU.
Note: Unity Pro v4.0 or higher is required for this functionality and it is only available on the embedded Ethernet port of the CPU.