Yes, street vendors in Italy sell hats. I bought one in Venice and wore it for my days in Italy. It never looked as good as it did before it was packed but it was fine to keep the sun off me in Italy. I finally threw it away after I got home because it was pretty mangled BUT it was only 10-15E so was not expecting much, lol!
Depending on when you are traveling, keep an eye on Costco if you are a member. I got a great straw hat there a couple of years ago. I won't buy a hat unless it has a chin strap as I live in a very windy area and this one had a strap and a very wide brim. I've packed it a few times but it is looking a bit battered and the brim has a fold in it that won't come out. I'll be watching to see if they have them again this year. It, too, was about $15.
You might also look at Dr Cynthia Bailey's website. She is a CA dermatologist and usually has good stuff. Her selection right now is not big - just one kind and it does not have a chin strap. Does have some nice colors but is fabric not straw.
You know we are going to want to know what you wind up choosing and how it works!