There is a flag for the United Federation of Planets, but it was only on-screen in one episode of the original series: "And the Children Shall Lead" (October 11, 1968).
Weirdly, the episode has nothing to do with the workings of the Federation, nor intergalactic politics, so it’s hard to say why the UFP flag was employed here.
In the episode "The Ultimate Computer" (March 8, 1968), the UFP pennant was painted on a Federation space station, but it was added digitally when the episode was remastered.