Is there any way to search for photos that don't have any tags?

Certainly. Please review the fields list of the search options for a complete list of standard search targets.

I'm not sure I follow your answer. I know how to search for a specific tag, and how to limit my searches to a limited number of fields (which can then be reduced in the search box for a particular search). What I am not clear on, is how do I search for NULL tags? My apologies if this is obvious.

What do you mean by searching for NULL tags? There is probably no such thing.

If you want to search for something which does not contain 'X' then you would do a search on !X (or NOT X). Search has full boolean logic

You pointed me in the right direction. A search on Tags using 'NOT " "' (NOT and a single space between quotes) will bring up untagged images. However, if you try to save this as a dynamic album, the search criteria is lost and needs to be reentered when the dynamic album is being created.