Here in Poland, I have no trouble buying both tartaric acid and potassium hydroxide. I'm having trouble sourcing cream of tartar though, so would it be enough to mix the acid and potassium hydroxide? Or do I need to get them to react in non-powder form and then dehydrate it back down to a powder for storage?
Potassium hydroxide is not something you'd want to eat. It's caustic (the base form of corrosive), and you ABSOLUTELY should not mix an anhydrous (pellet or powdered) form of it directly into food, whether or not you mix it with another powder first. Also, if you get the proportions off and the acid and base do not fully react, you could end up with a corrosive result. That's not something you want in food.
The chemical synthesis of potassium bitartrate (otherwise known as "cream of tartar") is relatively straightforward. Usually, you'd have a solution with a high concentration of tartaric acid, to which you'd add a solution of potassium hydroxide until the salt (potassium bitartrate) begins to precipitate out. You'd then continue to titrate to obtain maximum precipitate. Lastly, you'd filter out this precipitate and dry it to have powdered cream of tartar.
But I cannot advise you to attempt this on your own unless you have chemistry training. First, as another answer points out, industrial chemicals are not generally designed to be "food safe," and although they are relatively pure, the small impurities may not be something you'd want to ingest, depending on how they were manufactured.
Second -- and more important -- it's potentially DANGEROUS unless you know what you are doing. Read the MSDS sheet on potassium hydroxide, particularly the section on acute health effects. (Note that the health hazard rating is '3', which is on a scale of 0 to 4, with 4 being the most hazardous.) You should generally wear gloves and goggles when making a potassium hydroxide solution to begin with. And even if you safely handle the stuff, you need to ensure it is properly reacted and the resulting cream of tartar properly filtered and dried, or else you could be putting a caustic agent into your food.
Basically, the potential for injury is high if you don't know what you're doing and mess something up. I wouldn't recommend it for home food applications unless you're a trained chemist.
If you attempt this reaction you will want to purify your potassium hydroxide and tartaric acid. With a little research you can usually determine what other substances will be in your industrial grade products, i.e. anti caking agent in the tartaric acid or surfactants in the potash.
From there you can perform a recrystallization to remove some of the contaminates using the different solubility of each chemical. If they are similarly soluble then you may have to perform it several times to make it reagent grade. If you are lucky, the contaminates with be either highly soluble or not soluble at all. Check solubility for distilled water, solvents, alcohols, etc. Sometimes one product will not dissolve in methanol and your desired product does dissolve.
I would recommend you just put all the effort this would take into finding cream of tartar; it is not dangerous, yet tartaric acid is, well, an acid and potassium hydroxide is very dangerous and can cause severe burns. It doesn't make sense that those are easy to come by but not the cream of tartar. You did check the spice aisle at the grocery store, correct? Its sold there as a leavening agent for things like lemon meringue. Be safe and good luck.
In any case, make sure that the reagents you get are food and/or "internal use" medical grade. Anything else can have poisonous contaminants.
It is unlikely that the dry powders would react to any significant degree; they would only do so at the surface at most. Reactions involving dry powders usually depend on the powder particles being either mechanically getting pulverized finer and finer, or being thermally decomposed into gases or liquids, in both cases by energy released from the reaction itself. But I think you wanted baking powder, not gun powder?