Bayon plans to appeal, according to a notice filed Thursday.
The voicemails left for the two politicians in June 2018 included threats to "feed them lead" and warned that "you will pay," prosecutors said.
"You are taking ours. We are taking yours," the messages said, according to prosecutors.
Authorities said they found a loaded rifle, 150 rounds of rifle ammunition and books on bomb-making in Bayon's home and a storage garage. He lives in Grand Island, New York between Niagara Falls and Buffalo.
The messages arrived a year after Scalise, now the No 2 House GOP leader, was gravely wounded when a gunman attacked a Republican practice for a congressional baseball game. Scalise was hospitalised for more than a month after suffering shattered bones and internal organ damage; doctors said he had nearly died.
Capitol Police and other officers killed the gunman, who had nursed grievances against Republican President Donald Trump and his party.
McMorris Rodgers was also in the Republican House leadership at the time of Bayon's voicemails.
Scalise spokeswoman Lauren Fine said Thursday that the congressman is grateful to law enforcement and the courts "for ensuring that justice was served."
"There's no place for violence or threats in our political discourse, and today's sentencing affirmed that," Fine said.
A message was sent Thursday to a spokesman for McMorris Rodgers.
A federal jury in Buffalo convicted Bayon in August of retaliating against a public official and making interstate threats.
- AP