ALL antique/flea markets take place on the weekends, so you need to take a look at those taking place as soon as you arrive and when you are about to leave... you won't be finding any for during the week.
As for windor shutters, all companies that sell windows shutters take away the old ones they replace, they are likely the best place to contact to find out how to get access to them.
Fattorini - Infissi
Via Roma, 44
56034 Chianni Pisa
Il Paraspiffero By Ermetica Inox Di Russo Cirino
Via del Commercio Sud, 57
All-Fer Di Ragoni Floriano
Via Volterrana, 4
The terms in italian are "serramenti" and "infissi" ;-) Infissi are for windows, while serramenti refers to locks/safety in general... anyways, these type of shops do the shutters, modern ones are in aluminum.