If you define
"internet access" as "having TCP/IP", it might have been
BSD 4.1a, which included the crufty BBN TCP/IP code, and was released in
1982. ARPANET switched from NCP to TCP/IP on January 1, 1983 ("flag day").
I'm a bit surpirzed no one has mentioned UUCP.
I did, at some point.
Peace... Sridhar