He is the current performer of Kermit the Frog, Floyd Pepper, Robin the Frog, Crazy Harry, Camilla the Chicken, Dr. Julius Strangepork, Uncle Deadly, Pops, Lew Zealand, and Sweetums. On July 10, 2017, it was announced that he would be taking over for Steve Whitmire as Kermit the Frog, because of the latter leaving the Muppets.
He also performed 80s Robot in The Muppets, and Constantine in Muppets Most Wanted and made a guest voice appearance as Crumpet the Frog in Amphibia, which was done in-character as Kermit himself doing the voice.
Outside Disney, he has become familiar for his work as Caroll Spinney's understudy for Big Bird on Sesame Street, which he fully succeeded Spinney in the role in 2018 when Spinney officially retired. Also on Sesame Street, Matt started doing the puppetry for Jerry Nelson's characters, including Count von Count, in the early 2000s, but Nelson continued providing the voices until his passing in 2012, after which Vogel took over as the voices of those characters as well.