After we had tried to organize latex-free condoms in dozens of super markets and pharmacies, via couch surfing hosts and friends, and only found one box of latex-free condoms in eastern Turkey, we tried again to find some online.
And then we found Trendyol. It’s a big online meta shop and with this service you can pay with an international credit card, can create an account without a Turkish ID and have your orders sent to thousands of mini markets and other shops in Turkey.
I would strongly recommend checking that web service if you need to get something special in Turkey.