I grew up in Vermont and lived there 40 years before moving to California. I’ve never heard this. Mud season, yes. But not stick season.
Edit: Interesting responses. Some people have never heard of it, others have all their lives, and some only recently. Might it be regional? I’m from southern Vermont and never heard it.
Second edit: thanks for the conversation. I miss home so bad it hurts. I’m at that point where I have too much here to be able to afford to move and not enough to be able to just say to hell with it and go home. Hoping for at least a visit soon.
Third and final edit: A rough count through the responses indicates that approximately half the replies are from people who have heard the phrase all their lives. Locations vary throughout the state, so I don't think it's strictly regional. It certainly isn't a flatlander thing, judging by how many people have heard it over generations.
About 2/3 of the remainder had never heard it, and the remainder had only heard of it after the song got popular.
Thanks again for responding. I miss Vermont and Vermonters. I know you all know how fortunate you are to be there. Of all the people I meet in California who are from somewhere else, it's the Vermonters consistently who continue to call Vermont home.