I assume you have an Inter-Tel and VPU combination instead of something else.

From what I remember, one is unable to use authenticated SMTP to relay email out from this setup, which is not very likeable to most cloud setups. If your email is onprem, it'd just go through as everything is local and matches up and goes. But the cloud assumes authenticated SMTP and "modern" mail setups.

One solution (instead of a dedicated exchange server) would be a tiny unix server running sendmail or postfix to do the relay out to the cloud for you, but that would require a little bit of experience setting that up. Wouldn't take much.