The series has quite a few actually. The episodes "Her Chance To Dream", "The Old Man And The Sea Duck", "Stormy Weather", and "Plunder And Lightning" have a fair share of moving moments along with other episodes.
In the end of the third part of "Plunder And Lightning", seeing Kit cry when he stays with the pirates after Baloo, Rebecca, and Molly had escaped, and seeing Molly cry when Baloo takes off and flies away at the end of the episode.
Also in "Jolly Molly Christmas", the last half of the episode where Molly finds out that "Santa Claus" is actually Louie, and runs out of his club crying. Not to mention the ending of the episode is extremely moving (see the Heartwarming Moments entry for details.)
This particular episode was lucky enough to have an Animation Bump that help sell its emotional moments.
The episode "Stormy Weather" where Kit and Baloo have their argument about Kit becoming a daredevil has one of the most seriously played uses of the trope You're Not My Father used in western animation when Baloo tries to explain how dangerous his stunts are getting and Kit shouts "No, you're not my dad!" and jumps in his bed crying. This gets much Harsher in Hindsight when you consider that Kit's nickname for Baloo is "Papa Bear", and completely explains Baloo's heartbroken response afterward.
In "From Here To Machinery" when it seems Higher For Hire is going out of business, Baloo is left to explain to Molly he is leaving. Molly angrily punches at him, refusing to let him go before bursting into tears. Shortly following this, Rebecca explains she has to sell the Sea Duck to prevent becoming bankrupt. While Kit is outraged by this, Baloo, who throughout the show has been fiercely protective of his plane, merely replies with a depressed "Let it go, Lil' Britches, let it go...".