Yeah, there's no such thing as a "career Oiler".
Of the guys who won all 5 Cups, every one played for at least one other NHL team. The closest here would be Randy Gregg... but he played 28 NHL games for the Canucks in 1991-92 (and even when in Edmonton, he frequently left the team for spells to focus on his medical degree, so not really "career Oiler" when hockey was his second career).
Of the "magnificent seven" (who all started in 1979 and stayed in Edmonton through multiple Cups) --- Semenko, Hunter, Lumley, Gretzky, Lowe, Messier, Fogolin --- all played for three or more NHL clubs, except for Lowe and Fogolin, who played for "only" two.
Of the current crop of active Oilers, the best candidate for "career Oiler" is certainly Ryan Nugent-Hopkins. He's from Alberta, drafted by and played only for Edmonton, and last summer signed a lucrative long-term contract with the club. He's 29, so still lots of time to leave, but so far, he seems the safest bet.