The correct way to write 'The Morgans" on a sign? [duplicate]

You generally ignore that the underlying word has an irregular plural, so it would be "the Wolfs". (This is the same rule as for irregular words in compounds where they aren't the main noun: "mongooses" is the plural of "mongoose" even though "goose" has an irregular plural.)

It's up to the family concerned, and it doesn't have to be the same even betwen branches of the same family. In LOTR, Bilbo refers to a family as "the Proudfoots" and is interrupted by a Proudfoot saying "Proud*feet*!" It's not meant seriously, but it does indicate that people call themselves what they want to.