Lets take Sarek. He is the Vulcan ambassador. Most of the time, he represents Vulcan. In other cases he represents the whole of the federation.
As the Vulcan ambassador, he represents the interests of vulcan to other federation members, non federation members, and is involved sometimes with federation as a whole.
To me an ambassador, that isn't representing the federation as a whole, only represents there planet, or some form of coalition or sector. They don't go to Earth for any formal state business, that is for counsel/senate members to do.
So, lets say Vulcan has a border dispute with Andoria, is it federation business? Probably not at a low level. Sarek would go to Andoria, or vice versa, and negotiate. This happens all the time between states, water rights, borders, etc. that normally doesn't involve federal level, unless they can't hash out an agreement.
So, Sarek would represent the Vulcan States interests with other member states, and other non member states where Vulcan has an interest.
From Journey to babel, ambassadors may represent member states in some type of votes where negotiations may be needed. Maybe that's the way it was set up instead of a counsel/senate vote, it would be an ambassador vote at the place where the interest is.
Now as said, planetary ambassadors may represent the Federation as a whole in certain matters, like say negotiations with the Klingons. I suspect that the Federation has a pool of Federation ambassadors, like Robert Fox from A Taste of Armageddon but most of them are planetary representatives that pull double duty or people that have decades of experiance like Sarek.
The federation is a very Weak Federal Government type of system, it has to be, by the 24th century its a 1000 light years wide. Travel time, even subspace time would dictate that individual states take care of themselves, with help in the form of star fleet and other stuff from a federal level Federation. Think of Deneva, no ship has visited in a year, there taking care of themselves.
You would have counsel/senate members that are more or less permanently stationed on Earth as a planet/sector representative. Could Counsel members be called ambassadors? Maybe? Then each planet would have "ambassadors at large" roaming around the federation representing there interests.