Were either of these Federation members?
At a guess, I would say that the Denebians are Federation members, otherwise they would not care about the Vulcan patent. However, that death penalty for fraud is probably a local law, unrelated to the interstellar law of the Federation. Would the Federation extradite Mudd to Deneb 5?
Meh. From reviewing the dialog, it sounds like Deneb V's biggest concern was simply that they didn't have the exclusivity that Mudd led them to believe they would have. That lack of exclusivity would matter if they had intended to leverage it by doing business in Federation space. There's no evidence that they were acting on behalf of
Vulcan. Besides, even if they were, it could simply be because of treaty.
No need at all for Deneb V to be in the Federation from that.
I. Mudd wrote:
MUDD: I sold the Denebians all the rights to a Vulcan fuel synthesiser.
KIRK: And the Denebians contacted the Vulcans.
MUDD: How'd you know?
KIRK: That's what I would have done.
MUDD: It's typical police mentality. They've got no sense of humour. They arrested me.
MCCOY: Oh, I find that shocking.
MUDD: Worse than that. Do know what the penalty for fraud is on Deneb Five?
SPOCK: The guilty party has his choice. Death by electrocution, death by gas, death by phaser, death by hanging.
MUDD: The key word in your entire peroration, Mister Spock, was, death. Barbarians. Well, of course, I left.