Airborne and naval landing operations are fairly difficult to pull off correctly, requiring great amounts of prior planning and quick thinking under fire to pull off correctly. The Soviet Union in 1939 was neither equipped with the airborne and naval infantry units to successfully conduct said operations, nor was the military predisposed towards said operations.

The Soviet Union was an early experimenter with airborne troops, forming their first airborne brigades in the early 1930s. By 1939, the USSR had 6 airborne brigades (which combined parachute, glider, and airlanding groups), each (ideally) with 3000 men and 11 light tanks, plus artillery, mortars, and heavy machine guns. These troops were very well trained and had a high percentage of Komsomol (the Communist Party youth league) members. However, they were completely untested in actual combat. While we obviously cannot talk about their performance in the Winter War, we can look to their performance in the early phases of the Soviet-German War, which literally failed to get off the ground. The Soviet Union had minimal numbers of modified TB-3 bombers for paratrooper delivery, since none of the proposed assault-landing bomber aviation regiments were formed by the start of Operation Barbarossa. In a planned airlanding operation near Bobruisk on 28 June 1941, there were not enough planes to carry the 4th Airborne Corps into battle, and thus the operation was aborted. The 214th Airborne Brigade (a component of this corps) was forced to become an impromptu motorized raiding party and was encircled as the Red Army retreated eastward. The majority of its soldiers eventually fought back to friendly lines or joined growing partisan bands. Thus, the airborne forces of the USSR, despite their high motivation, had overall poor combat readiness, and could not truly be even considered paratrooper units because of their lack of planes.

As for naval infantry forces, they were basically nonexistent before the beginning of the Soviet-German War. A separate naval infantry brigade was formed in the Baltic Fleet in July 1939, but this was the only formation larger than a company that existed before July 1941. In a somewhat interesting parallel to the airborne forces, this naval infantry brigade was only in name only, since it was not trained in naval landings, nor were there any specialized ships to conduct said operations. Thus, the Red Army would have needed to build those had it wanted to use even non-specialized troops for a naval invasion of Finland. Again, we can look to the Soviet-German War for potential examples of how the naval operations might have proceeded by looking to the Red Army's unsuccessful counter-assaults on the Kerch Peninsula. Had the Soviet Union possessed the appropriate force and tools to conduct naval operations, there is still no guarantee it would have been successful.

Of course, any discussion of the Red Army in the early phases of the Second World War is incomplete without mentioning the effects of the Great Purge on army leadership. The Great Purge discouraged creativity and initiative in officers of all levels of command, leading officers to propose textbook solutions to non-textbook problems. Why risk a delicate, untested plan that requires exceptional coordination between multiple branches of service when its failure could potential mean imprisonment or death? Red Army leadership became overall less innovative because of the Purge. Furthermore, with specific regard to the airborne forces, many of the officers who championed their theoretical usage in deep battle operations, such as Tukhachevsky, had been killed or otherwise removed from command. While these theories continued to be circulated, there was an overall lack of acceptance and understanding of deep battle as an operational theory. Commanders did not understand potential applications of airborne and naval infantry forces in service of overall tactical and strategic goals, and thus did not apply them as solutions during the Winter War.

Sources:

Abramov, E. P. Черная смерть: Советская морская пехота в бою [Black Death: Soviet Naval Infantry at War]. Eksmo, 2009.

Glantz, David M. Stumbling Colossus: The Red Army on the Eve of World War. Univ. Press of Kansas, 1998.