I'd like to know where I can go to search old issues of popular
children's magazines.

I am looking for a specific issue, circa 1979, of a children's
magazine that contained short fiction stories. The magazine may have
been "Humpty Dumpty,"  "Jack and Jill," "Highlights,"  or similar. 
The issue I'm interested in had a story entitled "The Blue Lady."  The
magazine itself was smallish--probably around 5" x 7".

You may want to try the publishers--they may be able to look through
their archives for the story. Even better would be to find a contact
name at the company, the switchboard might not put you through to
anyone without a name.