If you ask “Where can I find any flea markets/vintage stores in Antalya?” for vintage shopping there are tree different places you may go:
Kaleici Old Town
This is the old town in Antalya and it is totally worth the visit. Easy to find such authentic places, friendly cats, traditional vintage shops and instagramable shots in the town.
And it is place to have a great walk with your family and to stop to have some tea in the local cafes.
Sobacılar ve Bakırcılar Çarşısı (Bazaar of stove producers and coppersmiths)
Sobacılar bazaar is place where mostly vintage Turkish handmade cupper tools and stoves are available
Region of Hans (Hanlar Bölgesi)
Region of Hanz in Turkish Hanlar Bölgesi is place for clothing. You may find vintage shoes or dress, etc.