I just finished reading a book that covers this exact topic, in part at least: Midnight at the Pera Palace: The Making of Modern Istanbul by Charles King (2015). It covers the city from about 1914 to 1950, but has a couple chapters devoted to the Second World War.

As noted Turkey was neutral in the war, as they had kept the close relations with Germany that dated back to the German Empire era (recall the Ottoman Empire fought with the Central Powers in the First World War). However the Turkish government also had good relations with the Soviet Union (who were one of the first states to recognize the Republic of Turkey in 1923, and provided a lot of support in that era), and did not want to fight a Soviet invasion of eastern Anatolia (which they did in the first war, which ended badly). This neutrality lasted until February 23, 1945, when the Turks finally declared war on Germany (it was a symbolic declaration, as the Turks didn't actually fight; it did allow them to join the UN right away though).

As a neutral city, Istanbul was favoured by both the Axis and Allied powers for various reasons. Espionage was quite frequent, as both sides sent people there. The location of Istanbul also made it a major site for Jews to escape: the border with Bulgaria (an Axis power) is something like 100 km or so (60 miles), and both Romania and Bulgaria were not as eager to enact the same anti-Semitic policies that Germany did (not to say they were not anti-Semitic there). A lot of ships tried to sail from Romania in particular and travel through the Bosporus Straight, which goes between Istanbul, and as such needed clearance from the Turkish authorities to do so. This was not something Turkey was eager to help with: as noted before they kept good relations with Germany, and feared that if it looked like they were helping Jews escape this would no longer be the case. That the British effectively barred Jewish immigration to Palestine during this time also made it difficult for people to escape, as the Turkish authorities required visas of the refugees destinations.

I can't say anything too specific about black market trading, but I would not be surprised if that was also happening there as well. The city was known for having some interesting characters, and as an outlet to travel between the Allied and Axis powers would have made a good spot to do it (and unlike Switzerland, it was not landlocked so easier for third-parties to reach). But I won't comment on anything specific.

One final comment on the book: King is a professor at Georgetown, and while most of his previous work is on Eastern Europe/the Caucasus, he's done a good job here of writing a popular history that shows the transformation of Istanbul in a really chaotic time, and really brings the story to life. Highly recommend it for further reading on the topic.