In general, because "bubble up" uses the word "up" it implies that the thing you are talking about is actually rising (due to bubbles). Therefore, it is usually used when talking about the gases or bubbles themselves rising to the surface (or sometimes when talking about something else being carried up to the surface by the motion of surrounding bubbles, etc).
Since these sorts of tablets are usually not carried up to the surface by the bubbles, therefore, that's not a term people would usually use in this case, I think. Instead, I would personally be inclined to say something like "vitamin C pills produce bubbles and dissolve when they are in the water" (but as others have mentioned, there are a number of possible other synonyms for "produce bubbles" which could be used as well)