https://www.kansaiscene.com/listings/category/flea-market/
http://freemarket-go.com/shousai_osaka_001.html (to be held at
Minato Machi River Place)
Unfortunately, it seems that most of the flea market events I found online happening ard Osaka/Kyoto are going to be outside of your May travel dates, but hopefuly the provided schedule information above would still be useful for others who will be travelling to Osaka/Kyoto area during the next couple weeks period of May.
Married people/couple travellers who are planning for their Osaka trip itinerary, if you happen to be base around Osaka during first week Fri of each month, I would prolly suggest you guys to drop by
Ohatsutenjin Shrine to pray for your current love life/other half as it is a famous lovers shrine shared by my own Osaka friends, after the shrine visit if you are interested then go check out the nearby flea market which is held there on every 1st week Fri of each month.
Don't forget to check out the
Sonezaki Ohatsutenjin-dori shopping street while on your way back to the station or just to grab a lunch/dinner meal.
How to get to Ohatsutenjin Shrine? You may alight at Umeda Station (Midosuji Line of Osaka subway), Higashi-Umeda station (Tanimachi Line of Osaka subway) or just at JR Osaka station and then it is a short 5-10 mins walk to the shrine temple. (Access map info can be found
here for those interested)
Hope the above helps.
