@The Efficacious Gentleman --
Magnesium hydroxide, Mg(OH)2, is the stuff I have purchased to use in deodorant.
Magnesium oxide, MgO, is hygroscopic, meaning it absorbs and reacts with water from the air -- or perspiration from your skin -- to form Mg(OH)2. That means you're going to end up with Mg(OH)2 no matter which one you start with as long as there's water present. That's why Milk of Magnesia can be produced by using either chemical.
To convert MgO powder into Mg(OH)2 powder, you would "slake" (mix) the MgO with water, pour off the free liquid, and then dry the sediment. That means if you can find MgO from local suppliers, but can't find Mg(OH)2, you could convert MgO into Mg(OH)2 if you wanted.
The only question I don't have an answer to is whether it would be wise to slake the MgO
before putting it in deo or if it's okay to use the MgO directly in the deo. At the moment I'd lean toward slaking first, because I can think of two potential problems that might happen if a person used MgO directly in deo --
First, if you used MgO directly in an anhydrous deo, I suspect the MgO at the surface of the product in your jar or tube might absorb water from the air. That might make the surface of the deo unpleasantly goopy.
Second, if you applied MgO directly to the skin, it is going to slake itself by reacting with water at the surface of the person's skin or their perspiration. It's possible that could be drying or irritating to the skin.