So I created a kit called Vote, Set 'hide' to true and the authlevel to 0.
I updated the config, Default.1 with kit "vote". When I do claim kit it tells me I've already redeemed the kit.

Do I need to use the  kit permission kit.vote?

Check if you have written the configuration correctly. Groups are usually written with all lowercase letters and also the default one. Example: 

"default.1": {
      "kit": "vote"
    }

Kits do not need permissions, just hide it and set authlevel to 1/2 to prevent other players using the kit, avoiding the vote. However, the kit is given directly when using the /vote command, just in case you have the full inventory and it is not received, then you can use the /claimkit command to receive all the missed kits.

This doesn't work, I voted. In chat typed /vote and recieved "Thank you for voted on Rust-Server...". No kit.
Tried /claimkit and got the message "You have already claimed the kit. Rewards config below:

{
  "Groups": {
    "default.1": {
      "kit": "vote"
    },
    "default.2": {},
    "vip.1": {}
  },
  "TopVoters": {
    "1": {
      "money": "1000",
      "rp": "100",
      "kit": ""
    },
    "2": {},
    "3": {}
  }
}​

The only explanation then is that you are not in the default group.

I'm set to the Admin group, but doesn't that mean you also get default group permissions.
I will try it with a few more people.

A player can be added to multiple groups, in fact when I vote on my server it works.