What food dissolves in acid but not in water?
If your question is specific about acid: Nothing. 

The conventional way acids work is by dissolving them in water. I even have some vague memories that it's not technically correct to call a substance an "acid" before it has been dissolved in water - for example, pure HCl (a gas) is not an acid, but once it is dissolved in water, it becomes hydrochlorid acid. pH, the well-known measure for acidity, is defined as the logarithm of the number of protons swimming around in the aqueous solution - and before you dissolve the acidic substance, the protons stay firmly attached to their host molecule. I have never heard of an acid working without water being present. 

I am not a chemist, and maybe there is an way to get acid to work using some other polar solvent instead of water. So I don't want to tell you that it's absolutely impossible to have an acid without water (although this could be the case, I just don't know). But: 


The acid itself is by definition water-soluble. You can't bring acid and water two together and not get a new, weaker, acid-water solution. 
I can't think of any edible polar solvent besides water. Ethanol could qualify, but most people don't consider 96.4% ethanol to be "edible". 


If it doesn't matter to you what the other liquid is (not an acid), you can easily use anything non-polar. In food, any fat will do. 

You could start by floating solid-fat spheres in water and then pouring them into oil. The problem here is that they will take quite a long time to dissolve (although this can be circumvented by using warm oil). The other way round will work better. Take something which dissolves in water quickly, and make the hollow sphere out of it. Float it in oil. Then you'll pour it into the water to release the fragrance. 

Crystals are best for dissolving rapidly in water. I think that either salt or sugar will work well for your spheres. Of course, you are left with two problems: 1) how to manufacture the hollow spheres, and 2) what fragrance to use which will not dissolve the spheres themselves. For 2), you are probably looking at essential oils (warning, use foodsafe ones, not all are!). For 1), you probably need some precision candy-making techniques. 



Update: letting food react with acid.

The answer above assumed that you are really looking for stuff which will dissolve in acid. The commenters pointed out that this is not necessary, as you can let the food react with the acid instead. It's an interesting idea, but it won't be easy to get done. 


From the major food groups, you can exclude all proteins. We are made of protein, and any acid strong enough to quickly corrode spheres of gelatin or similar will ulcerate our mouths right away. 
You can also exclude the sugars. They are water-soluble. 
The polysaccharides are a better bet. Especially the hemicelluloses should be good candidates, as they are insoluble in water, and the result of the hydrolisis is edible (sugars and some polisaccharides). But they typically need very strong acids for the reaction. You might try to use a concentrated edible acid for the reaction, then dilute the result for consumption, but you'll have to pay lots of attention to proper ratio. Also, I don't know how you'll manage to physically shape one of the insoluble polysaccharides into a water-tight capsule at home. 
The fats are also bad candidates. They are largely unimpressed by acids, in fact their most interesting reactions are with bases (saponification, etc.) The short unsaturated ones will probably react with an acid, but you'll need a fairly strong one, and the result won't really disappear. 
The probably easiest way to do this is with salts. Especially the sodium salts tend to be edible and to react readily with acids. Plain old calcium carbonate sounds like a promising material, and with a bit of heating you can get it to react very quickly with acid solutions dilute enough to drink afterwards. Again, you have the problem of somehow making spheres out of it.