Japan did have naval forces at the time, and they probably fought the Mongolians a few times.
The samurai Takezaki Suenaga, a gokenin from Higo in central Kyūshū, was a veteran of both wars. To showcase his valour in battle (to request rewards from the government), Takezaki commissioned the Mōko Shūrai Ekotoba, an illustrated account of the Mongol invasions. Notably it contained several depictions of Japanese ships and naval battles.
While some fighting at sea most likely did occur, there's a paucity of records on the actual details. Western sources have generally dismissed these as little more than harassment tactics, if even mentioning them at all. Any battle that took place were probably quite minor.
Nonetheless, during the second invasion two relatively more notable engagements occurred.
Sea Battle of Takashima-oki.
After linking up at Hiradoshima, the Mongol fleets advanced to Takashima in preparation for assaulting Daizaifu, the Japanese HQ. On 27 July (in the traditional calendar), a Japanese fleet attacked the moored ships off the shores of Takashima, and supposedly fought till nightfall.
Battle of Mikuriya Sea.
The famous typhoon that wrecked the Mongol ships occurred a couple of days later, on 30 July. The Japanese fleet launched to clear the surviving Mongols from Hakata Bay. At this point the Mongol fleet was either retreating, or the commanders had taken the best ships and fled first. This was probably a pursuit action on the part of the Japanese.
There was also some mentions of the navy during first invasion, but I don't think it saw any action. According to the
Rekidai Kouki, after the Mongols attacked Tsushima and Iki, the Dazaifu sent a fleet of 300 ships. They probably pursued the retreating Mongolian fleet and came across some shipwrecks from a typhoon.
《歴代皇紀》 文永十一年十月五日、蒙古賊船着岸對馬壹岐攻二島土民、廿日、大宰府以三百餘艘之兵船發向、 賊船二百餘艘漂蕩、神威力云々