There are a couple of approaches I can think of to archive outbound SMS messages sent from your RingCentral extension's phone numbers into Gmail. Nearly all of these ideas require development of a custom application using the RingCentral Connect Platform.
1. Application
Subscribes (uses RingCentral PushNotification API resource) to the
presence data for your extensions you need to monitor/archive in your RingCentral Account. Filter the events for outbound SMS messages, and capture the messages, and send to Gmail via the
Gmail API.
2. Application to create Webhooks which POST to web server and relays to Zapier, where you create a ZAP which sends the Gmail from the Webhook data.
There are of course other ways to solve this problem, but these are the first two that come to mind.
Thank you, Benjamin, for the ideas. I was really hoping not to have to go the custom application (we are a small company, with limited tech dollars). I thought there may be an "off the shelf" tool (aside from Zapier) that other companies use to accomplish this function. If there are any other ways you (or other users) can suggest, I'd be grateful.
We currently do not have an "off the shelf" tool for accomplishing this precise goal, so I am afraid that a custom application might be the only path forward at this time. If you are in immediate need of this functionality, you could contact your RingCentral Account Executive and ask them about the RingCentral Development Professional Services.
