Is it possible to hide the Member Search tab?

You would need to disable access to profiles or hide it via CSS. There really isn't another option at this time.

@Ehren is there a good way to hide it via CSS? My attempts leave a little arrow button remaining.

I'm now using this and it seems to work.

#ipsTabs_search_searchMembers { display: none; }
#ipsTabs_search .ipsTabs__scrollers { display: none; }

Yep, that's correct. You can also comma separate the selectors if they share the same properties:

#ipsTabs_search_searchMembers,
#ipsTabs_search .ipsTabs__scrollers {
	display: none;
}