Yup and it's the same in France; it's a market 'aux puces' ( with fleas ) there too. Much of what they sold was old clothing. Brewer's Dictionary of Phrase and Fable says "the clutter of bric a brac and old clothes was conducive to fleas".
I found a French site who claims that the term started in St-Ouen, France, around 1880.Like any popular term it catches fast,or not at all,and it seems that, from mouth to ear the term was soon so "catchy" that the French reporters started to use it......... "et voilà"
There is a book out called "Les puces ont cent ans" meaning "The fleas are one hundred years old" written by Jean Bedel, unfortunetly the book and the site are in French only.
If there were ever a true 'flea' market it should have been Club Row Market, in London's Shoreditch. It was for pets; you could buy pups, kittens, canaries, parrots or most anything else small that was furred or feathered there . Now it sells furniture and general goods; the animals market stopped about thirty years ago because of complaints about animal welfare, puppy farmers and so on.