My mother and I are learning about the Mongol siege of Caffa/Kaffa(?) and how people could escape the siege by boat because Caffa was a sea town, and we were wondering why the Mongols didn’t just...build a boat or two and besiege them from the seaside? And that got us wondering, did the Mongols ever have a navy, or use the navy of conquered peoples?
Yes, most famously the fleet with which Kublai Khan attempted to invade Japan in the 13th century. He is at this point the Emperor of the new Yuan dynasty in China. The invasion of Japan is (by some accounts) at the instigation and with the assistance of the King of Korea, so you can quibble over whether this is a "Mongol Navy", a "Korean Navy" or a "Chinese Navy", but it's a distinction without much of a difference.
The Yuan-shi (the history of the Yuan dynasty) isn't terribly reliable with numbers, but it's safe to say that we're speaking of hundreds of ships and many thousands of fighting men transported. This naval capacity was never an "organic" capability of the Mongols themselves-- they were steppe peoples. Kublai had no problem ordering the Chinese to prepare an invasion fleet for him, but he didn't have his own "Mongol fleet".
As for your last suggestion (the navy of conquered people), I once posted a brief summary of the composition of the 'Mongol fleets' that invaded Western Japan respectively in 1274 and in 1281, in the following question thread: Why did the Mongols decide to attack the Hakata Bay area during their attempted invasions of Japan in 1274 and 1281?