Democrat is a noun. Democratic is an adjective. You can say "vote for a Democrat," but you can't say "vote Democrat" because you're referring to the Democratic Party where "Democratic" is the adjective. In that case, it would be "vote Democratic."
Republicans love to use slurs like "the Democrat Party." That's probably where you've heard it.
This is just grammar nazi bullshit. Give me empirical evidence that anybody would say "Vote Democratic". It's a phrase I've never heard but I'm not American and not a native speaker. The authority lies with native speakers but not language purists who think they are better than others.
Not trying to be better than anyone, and I do agree that it may not be the biggest of deals. But I'm a native speaker, and I'm just pointing out the grammar part and making people aware that it's a Republican epithet.
First: I'm sorry for having mistaken you for a jerk. There have been a lot over on r*ddit and here I have seen them too, but you aren't one of them and I'm sorry.
Second: I'm still not convinced either. It doesn't say "Vote the Democrat party", I see now how this would be wrong. Adjectives always stay before a noun (or at the end of a sentence after a form of to be). In the case of "Vote Democrats", "vote" is the verb and "Democrats" the object and therefore a noun (in contrast to "democratic votes" were votes is a noun and democratic the adjective).
Sure, the post doesn't say that neither. "Vote a Democrat" or "vote your local Democrat" would work but a singular without an article of any kind not so much.