Well, that's gonna depend on what the adjective modifying the noun is gonna be, isn't it?
When you say "Federation ambassador," do you mean "Ambassador of the United Federation of Planets to X," or do you mean, "Ambassador of Federation Member X to the United Federation of Planets," or do you mean, "Ambassador of Non-Federation State X to the United Federation of Planets?" Because that's gonna tell you who the ambassador works for.
Like, I think we can all agree that the ambassador of a foreign state to the Federation has no authority over Starfleet. Similarly, I have a lot of trouble imagining that the ambassador of a Federation Member State to the Federation would have authority over Starfleet, because ultimately, that ambassador does not work for the Federation -- they work for the individual Federation Member rather than the UFP as a whole.
To me, it makes far more sense to assume that only ambassadors of the Federation itself would have any legal authority over Starfleet, in specific circumstances. And I think it makes sense that in an interstellar state where real-time or near-real-time communication with the Federation government is not always possible, the Federation might legally empower some of its ambassadors to make certain foreign policy decisions on behalf of the Federation in specific circumstances.