Limited to political activity within a given country (i.e. not manipulating another country) this is a somewhat strange question. What do you mean by "support" and what do you think "democracy" means?
support You can advocate, peacefully, to worship Satan, as long your advocating does not break any laws of the country. There is no need to "support" you. The only obligation of a democracy is not to suppress your legal political expression.
democracy. This means rule by the people. Once more than 50% of the voters agree with you, well, you are the democratic majority. Until then, you would just be another party with some agenda that other people don't agree with. No more, no less.
limits on democracy. Most well-run democratic countries have some form of formally-guaranteed rights, looked after by high court(s), that limit what laws even a democratically-elected government can put in place and what actions that government can take.
For example, Canada's prohibition on gay marriages first got annulled by
an Ontario Court. Brown vs Board of Education decided
separate but equal wasn't equal. These decisions struck down laws originally passed by a democratically-elected government.
So, in your example, who cares about Sharia law? If, and that is a big if, Sharia Law did not violate any guaranteed rights and existing laws, then in theory, it could be adopted if a majority of voters thought it was good idea, making it, by definition, democratic.
Of course, common sense didn't stop state lawmakers in a number of US states (least likely to be very Muslim) to pass grandstanding state laws forbidding Sharia law.
The US also strenuously, especially in the 1950s, suppressed Communism, using all sorts of over the top and not very democratic means, during the McCarthy period. But Communism on its own never got much appeal so that wasn't really needed. (note that watching out for espionage by Communist sympathizers in security sensitive positions is something entirely different).
Communist governments don't have a very good track record of allowing voters to decide to switch back to a non-Communist government through free elections. So that could make their election a somewhat special case.
Last but not least, though I don't know the exact details, Hitler managed to transform a sizable, but still minority, vote, using dubious means to declare a state of emergency and then essentially abolish democracy.
So, under certain conditions - cough Jan 6th cough - democracies can stop functioning as democracies. But in normal conditions your question doesn't make that much sense.