There will always be people gaming the system if it suits them. Moreover, if you make it such that you don't need cookies at all you'd be open to very simple attacks.
I think you'll want to consider ways to increase the economic cost of users operating under a cloud of suspicion.
For example, if a user with the same cookie tries to re-submit the vote, that can obviously be stopped easily.
If a user with a different cookie but from the same IP does the same thing, it could be coming from a proxy/firewall so you may want to be cautious and force them to do something extra, like a simple CAPTCHA. Once they've done this, if they behave properly nothing new is required as long as their new cookie stays with them.
This implies that people without cookies can still participate, but they have to re-enter the letter sequence or whatever each time. A hassle, but they're likely used to sites not working without cookies. They'd be able to make an exception if required.
You really won't be able to deal with users sitting over a pool of IPs (real or otherwise) and exploiting new and dynamic attack vectors on your site. In that case, their economic investment will be more than yours and, frankly, you'll lose. At that point, you're just competing to maintain the rules of your system. That's when you should explore requiring signup/email/mobile/SMS confirmation to up the ante.