Hi Jo, sorry if this is a bit long but I thought it was a really interesting question!
As fair as I’ve been able to find out, it’s ultimately one of those things where N/A or n/a tends to be more common, so style guides just lock it in for consistency.
Anecdotally, I’ve noticed that noun acronyms tend not to use a slash (and are always capitalised) while adjective and other acronyms tend to use a slash (and may or may not be capitalised). For example CO means “commanding officer”, but c/o means “care of”.
Personally I think NA for not applicable has gained popularity due to it’s common usage in computerised forms - plenty of poorly designed forms have required fields where they shouldn’t, and back in the day it wasn’t uncommon for forms to only accept alphanumeric characters. Plus it’s a little faster to write. However a slash makes it that tiny bit more obvious that it’s not, for example, just an unusual name.
I’ve rarely seen not applicable abbreviated to n.a. which may simply be regional, but I suspect is more likely due to periods in acronyms also being associated with the ‘noun type’, just a little old fashioned now. There’s definitely a trend, at least in government writing, to avoid periods in acronyms entirely.