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.