We have heard of two voting processes in the Federation so far. In "Homefront", UFP President Jaresh-Inyo has this to say:
JARESH-INYO: I never sought this job. I was content to simply represent my people on the Federation Council. When they asked me to submit my name for election, I almost said no. Today I wish I had.
This describes two separate things: that J-I used to represent his people (we have no idea who they were - his species, the people from his home planet, the people from his political party, a random flock assigned him?), and that he later underwent an election to become the UFP Prez.
The other process is from "Journey to Babel", where Ambassadors journey to Babel on an apparent fact-finding mission, after which there will be a vote, and it appears to be divided by either planet, or species, or culture. That is, the Tellarite rep presses Sarek for revealing "the Vulcan vote". And "Vulcan" may mean any of the above three.
Neither of the above is a popular vote unless we want to believe it is. One means voting on a person, whose political powers (beyond the ability to declare martial law on Earth) are unknown. The other means voting directly on an issue. We don't know who is casting the votes - the people, agents elected by the people, or agents appointed without regard to the people. We don't even know the name of the forum at which the voting for the President takes place; in the TOS episode, the vote on Coridan is at "the interplanetary conference", either a proper name or a mere description for the forum.
Apart from the above two bits on elections, there's this from "Rapture" as regards the process of becoming a UFP member, from Adm. Whatley:
Now, Bajor's admission is only the beginning. Now comes the hard part. Federation council members have to be chosen...
No idea who does the choosing - some Bajoran organization, democratic or theocratic or other, or perhaps some UFP board, democratic or bureaucratic or whatever? No idea what those members do, either, but the plural is interesting in at least slightly detracting from the idea that every planet/culture/species would have a sole representative or a single vote.
On such flimsy evidence, it's difficult to argue for or against the existence of political parties, or specieist parties, or planetary parties. We never hear anybody campaign for getting elected. We never hear anybody express support for or opposition to a candidate. Nobody ever votes on screen, or tells of voting.
So at best we can go by the novels. And they don't do anything fancy: the UFP is essentially a generic Western democracy in space, with many details gleaned from the US model specifically.
Timo Saloniemi