In NPCs, the only paladin we see become one is directly, divinely appointed. But that would throw most games off. (Besides, he's an NPC.0:)
And in Order of the Stick, though they even know they are in a game, the paladins who've taken center stage have belonged to an order with a mission. (And the far-faring paladin is one they send on missions.)
Even in Rusty and Co., Madeline the Paladin goes on missions from some kind of church -- since Derek the Cleric gives her prophecies to direct her.
The random adventurer paladin would really interesting to world-build, even if possible.