Welcome to community @user333​ , as far as I know, It's not possible to forward the conversation to e-mail. 

What about add these team members to Intercom, Assign them conversation and they will get an e-mail? They also can reply from this e-mail = )

That will be much easier for your team, they will just assign conversation instead of forward it. 

You can add team members and don't assign them Inbox license, That means that you will not pay for extra. 

* They only can reply assigned conversations via e-mail or via the Intercom app (But they will don't have access to Inbox and assignment roles (Only Manual assignment)).

Thank you Roy - very interesting. We are a Education Institute so about 400plus employees. I noticed that you can add the Outlook Calendar App and was wondering if it would be the same that you would have to add an Outlook email app/extension to get more email functions... ?

@user333​ , There no such extension.

You can also use Slack integration, transfer the incoming conversation to the slack channel, and allow your 400+ Employees to answer them. 

If you are interested in my opinion, better if customer support answers to customers. This support team can have contact with internal employees via Slack or Outlook chat. In this way, customer support will be more quality-oriented, instead of giving 400+ people able to answer customers :/

More about Slack integration: https://www.intercom.com/help/en/articles/1501873-slack-app/p>

You can also add employees to Intercom but they can view your contact list and all outgoing chats