How can I make it so they can visit their dashboard or a page on my website to access their gmail emails and send emails out? Can this be done via API?
In short, i'd like to keep them on the site rather then have them leave to check emails.
Out of the box, no. There are plugins that try to implement mail clients, such as WP SimpleMail, but I cannot guarantee that it will work, or that it will work well.
Building this yourself would be a massive piece of work, and easily several months of development work, if not a year
Keep in mind that the Google APIs are offtopic here, as are plugin recommendations
Your options are:
- send them to gmail
- test and install plugins that implement basic email clients and hope they work well
- spend 6 months building your own email client in PHP from scratch