The obvious answer is yes, the War of 1812. Then you have Germany, in 1914, governed by a Reichstag, although not to the same degree the Weimar Republic was. Also in WW I, Italy fought for the Central powers at the beginning of the war . . . against France. I'd have to check the actual status of France under Napoleon, but I don't think it was technically the "empire" until 1803, meaning England declared war on France sometime while it was still under the Directory (i.e., a Republic).
Japan also was ostensibly governed by the Diet, with the Emperor Hirohito merely a figurehead politically, but whether Japan actually declared war on China, or just invaded, I don't know.
well, no, because I don't know hence the question ... it's been interesting to read what definitions of "democracy" are