Currently, if you search order:character or order:species, you get them primarily ordered by the number of tags in these categories. I’m wondering if there’s a way to search alphabetically. For example, any post with an aardvark tag gets grouped in a search with other posts including aardvarks when performing a search with the order:species tag.

That is not possible, no, and there is no simple change to the site that can make it possible

In the documents used for searching posts, they have no idea what the category of their individual tags is, only how many they have of each category