Anyone can be a Hunter.
Nearly every race has valid Hunter class lore, each with its own twist on it. For example:
But what’s the coolest one of all? Which race feels the most appropriate for the Hunter class, looks best in its transmog, and fits with its pet-based playstyle, weapons, and abilities?
It will forever be dwarf for me thanks to the original opening cinematic with the bear in the hills above the gates of Ironforge.
Gotta agree here. There is just something so iconic about the Dwarf Hunters with their Boomstick.
Dwarf or Elf.
Dwarf, Tauren, Night Elf.
Adore a Night Elf Hunter but as I mained Horde my Troll is also awesome.
Well I like my goblin sv and the explosive shot and wildfire bomb theme. Wish there were spec or class that played more around this explosives and gadgets type of theme.
Elven rangers hard vote… OR! you can go like a Rexxar orc beastmaster hunter.
But yeah… the Elven rangers have got the most spotlight and awesomeness in this game so far.
Dwarves get a honorable mention as well. The first hunter we ever saw was a Dwarf hunter in the first classic cinematic with his bear and blunderbuss rifle!
I love the night elf he is BadFarter
Trolls would have been my total favorite, if Blizzard hadn’t gotten rid of Throwing weapons. I would have loved to play a Troll Hunter with Throwing Spear or Throwing Axes 
honestly
Human is the coolest.
but the night elf racial is BiS for hunter to go into stealth and lie in wait in BG’s
Trolls for Horde, Dwarves for Alliance.
Worgen for Aliance and Vulpera and/or Elf for the Horde
Coolest race is draenei and their hunters are Rangari who were scouts all across Draenor, including the jungles of Tanaan.
Tauren easy choice,
The first character I made was a Tauren Hunter and with the release of classic I started back with the same name and appearance
Pandaren 10char
Completely dependent on the ‘fantasy’ or ‘theme’ you’re going for (and of course it’s all subjective in the end).
For an ‘explorer/adventurer’ type hunter, I think a human is best.
Example:
Although I also like that theme on gnomes.
Example:
For a feral hunter clad in fur and wearing a wolf pelt; I think orc is best.
For a drust inspired, creepy type of hunter, I think forsaken is best.
Example:
Etc. It all depends. 
Ps: I want the ability to transfer my ‘timerunners’ to the regular game. I want to start taming other pets besides the stuff you can find in Pandaria.
The answer is always Dark Irons. Racial cleanse is unbeatable, plus it’s not only a cleanse but a fancy steroid. Plus indoor movement for farming. Plus mole machines. Plus…it’s dwarves with fire beards, come on. Perfect gunners, great gun animations but also works with bows.
If you’re on elven side, then it’s Void Elves. More fancy and emo than OG Blood Elves, plus the whole aesthetics of Rangers. Plus Alleria and Void will be core themes for next bunch of expansions. Though animations are lacking – they look absolutely goofy when sprinting, they only look good with bows and they sit on the drakes like slingshots. But hey, tmog discounts!
Same, I started this game as a Dwarf Hunter. I play Horde now, but I make a point out of it to level him to cap whenever I play <3
He will forever run around in his full Giantstalker set, AV Tabard, and Blastershot Launcher. <3
That being said I leveled a Vulpera Hunter on Remix and I really like him. Hunter/Rogue are the most fitting classes for Vulpera I think.
My first thought aswell, aside from the obvious Dwarven Hunter with bear (opening cinematic) or Nelf