You're not going to be able to set a web based mail client for this action without some sort of 3rd party plug in (I'm not aware of any that do this). You could potentially create a VBA (visual basic for applications) script that might help.
Maybe one of these will give you some ideas or help you out in alternative approaches.
Why not use Outlook?
If you have Office Pro Plus then you do have Outlook available and you can configure it to work with your Gmail account, which you then could leverage to get this to attach in Outlook.
Google Drive Plug In
I believe there is a Google Drive Plugin for Office that you may be able to leverage to save to Drive directly. That should at least lower the friction in getting access from Gmail.
Use the Office 365 Suite
If you want a more integrated experience - I would look at using Office 365 suite. If you are not already - Microsoft offers free subscriptions to non-profits for Online services, if they qualify. Link to MS Non-Profit Page for O365. If you're not signed up yet - I recommend it you can get greatly discounted rate on Office 365 Subscriptions like Office Pro Plus. Note - you don't necessarily have to "migrate" to Office 365 to get this done. While that would be the best long term strategy if you want to go that route, you can just have mail forwarded to the Office Online address to test it out. I'm not sure this solution will give you much more (if anything)form a user experience standpoint over Gmail/Google Docs plugins if you are not using Outlook directly. It might actually be a big negative if your users don't like the Outlook layout.
Try one of the Integration Services
One other thing I can think of that you may be able to leverage is something like Zapier. There is already a "Zap" out there to send an email message when a new file is added to OneDrive. You may be able to tweak this or create your own to get close to what you want to do.
Very valid points. I would also like to add that you couod just connect Outlook to Gmail, allowing you to send gmail from within Outlook. Unless you have a specific reason for not doing so, this is a viable solution – InterLinked – 2016-04-08T23:16:50.510