One thing that would allow it but that we don't support yet would be webhooks — when a purchase would get made, an HTTP POST request would be sent to a server of your choice with purchase info, letting you send the e-mail. We don't support that (yet).
A solution you can do right now is to encourage your users to play your game through the itch.io app, and use an app manifest to request an API token — sending this to your server will let you retrieve purchase information for the player, generate discord link for them and let them join the discord server from clicking a button in your game: a pretty low-friction approach if you ask me! (no need for them to open their e-mail at all).
Let me know if you have further thoughts — webhooks have been on my mind for a while but I can't guarantee an ETA for the feature, whereas manifests are working right now.