On UESP, I'm reading a lot about how "apothecary merchants" have a certain percent chance of having a certain ingredient for sale.

What are "apothecary" merchants, and who are some of the best ones?

They're alchemists. (http://www.uesp.net/wiki/Category:Skyrim-Apothecary)

If a merchant doesn't have the ingredient I'm looking for, how long do I need to wait in order to reset their inventory?

And... do I have to step outside the shop while I wait?

42 hours, I think. Also, you can try resurrecting them if you're playing on PC.

It takes two days for a shops inventory to reset,and yes you do have to be out of the shop.

When is a shop's inventory calculated? Maybe it would be easier if I saved immediately before this point and reloaded if they don't have what I need.

I'm not sure but I would think that as long as you haven't entered the shop within the last two days that the inventory would be set as soon as you do enter the shop.
So saving then trying again if they didn't have what you needed would be the fastest way.