No, not that kind. I'm talking about fawn as in fawning, being obsequious, or as Peter Rozovsky said in the comment field here recently, being a lickspittle--a great word for which I praise whoever first came up with it. But that is not our quest today.
Fawning is also a good word, but when you try and reach back in time for the sense of it, well, it's difficult to come up with the connection. Fawns may have some negative traits...oh, who am I kidding--fawns have no negative traits, none whatsoever, unless it's that they grow up to become deers that eat a lot of your favorite plants. Remember The Yearling? No, try not to. It's sad.
Anyway, fawns may be timid or shy, but they aren't exactly known for kissing up to people. So where does this word come from?
It turns out that there are two kinds of fawns. The little furry hoofed kind, which comes from the Old French faon or feon, meaning "young animal". It goes all the way back to the Latin fetus which I probably don't have to translate for you. The Online Etymology Dictionary tells us that originally fawn meant any young animal, and was "Still used of the young of any animal in King James I's private translation of the Psalms." but mostly just referred to young deer from around the fifteenth century.
The other kind of fawn, as in fawning, has nothing to do with all that. It goes back to fægnian, which is an Old English word meaning "to be glad, to rejoice, exult." It's connected to that old-fashioned sounding word fain, as Gonzago uses it in The Tempest, when he says, "I would fain die a dry death.", fain here meaning gladly. None of this sounds much like fawning, though, does it? Well, the Online Etymology dictionary tells us that in Middle English, fawn was used to refer to expressions of delight, such as a dog's tail wagging. Somehow this perfectly lovely thing that dogs do changes from a good thing in the early 13th century to a bad thing in the early fourteenth, when it starts to take on its present meaning of groveling or acting slavishly. In other words it amounts to a smear job on dogs. And actually, fawns.