Is there any way to add Email API into Forge Application? I need to be able to send emails to users? If the Atlassian Email API is not supported, is it possible to add another 3rd party solution or will this be blocked by Forge Application?
I think the Forge Remote is the way you look for. If you need to communicate with various and dynamically specified 3rd-party services (i.e. whose URLs are not known prior to app installation, and therefore cannot be indicated in the app manifest) - such as a mail server - you should set up a (trusted) backend service to do this job for you. All such kind of workloads can be outsourced (in the future) in this “semi-transparent” model.
If you are writing a JIRA app, have you considered: https://developer.atlassian.com/cloud/jira/platform/rest/v3/api-group-issues/#api-rest-api-3-issue-issueidorkey-notify-post ?