The point of most captchas is to stop someone from writing a program that abuses your system. If you're not doing that, what exactly are you "verifying"?
For another website, I used to store the IP address and do checks but for a simple system that seem to be overkill.
All of the websites have no user accounts.
Is there another way to do this? Where you want some voting to happen but with no user accounts. And you want it to be dirt simple.
If you start with the assumption that no one will try to attack your site, then you don't need a captcha at all. You can give the user any annoying/time-consuming task (e.g. "click a square ten times"), or you can insert an inconvenient delay in vote submission. However, people may object less to the inconvenience of something that looks like a captcha, because they are used to them.
Needless to say, almost no one uses captchas for this purpose. Instead, they use captchas to make their sites more resistant to automated abuse -- as you almost certainly will need if your sites become popular.
The site dos not take in user generated content that is immediately displayed. The data is first curated and verified so the attack will be a bit harder. So there is fresh (delayed) content constantly.
That is exactly the reason of the captcha: to give a person something to do that wastes a bit of their time and that is familiar to them.
Yes, in other sites I have used captchas to prevent automated abuse.