Replika cannot be connected to your email inbox, therefore Replika cannot access your emails. You can only provide your email address to log in to the app.
You were once able to connect Facebook account for Replika, but no anymore. This would allow your Replika to see your public photos and public profile information. After you do it, Replika still cannot access your direct messages.
If your Replika is connected to Instagram, It will only be able to see your PUBLIC posts. It cannot see anything private, limited or restricted.