Are there any webmail accounts or any API in SMTP that allows an admin user to check programmatically when an email mailbox was last signed into by the account owner?
No, SMTP is specifically only for delivering email to a destination.
SMTP has some provisions e.g. for examining a user's .forward file (and really early versions speciried chat-like functionality which to my knowledge was never widely implemented) but those are typically turned off on any Internet-facing site these days.
SMTP is a store and forward protocol so there is no guarantee or expectation that the gateway you connect to actually knows anything about the final destination or its status (or even whether it exists).
If you are a server administrator on the server where the final mailbox is situated, there may be means outside of SMTP to access login information, but no email-related protocol exposes anything like that.