The accordion is used all the time in klezmer these days, as it can cover many (all?!) roles. There is one accordion player of note in the annals of North American klezmer history - Mishka Ziganoff, who was probably the most-recorded accordion player on the klezmer scene.
The accordion can be seen and heard as very much a part of Argentinian klezmer bands in the 1950s.
The accordion was sometimes known as harmoshke.
Klezmer accordionists in the late 19th and earlier 20th centuries included:
Argentina Amalia Dalle Mulle ('Mima'), Iosele Grinblat (also clarinet)
Poland Poldek Koslowski
USA Mishka Ziganoff, Seymour "Cy Mann" Megenheimer
Klezmer accordionists in the later 20th century and today include:
Argentina Cesar Lerner
Canada Josh Dolgin (also piano)
Finland Eva Jacob
France Alexis Kune, Jérome Block
Germany Alan Bern (also piano), Patrick Farrell (also piano), Sanne Möricke, Szilvia Csaranko (also piano)
Israel Emil Kroiter
New Zealand Ross Harris
Sweden Miriam Oldenberg
UK Carol Isaacs, Gabriel Ellenberg, Jim Marcovitch, Josh Middleton, Michael Alpert
USA Christina Crowder, Ilya Shneyveys (also piano), Ira Khonen Temple, Josh Horowitz (also piano, tsimbl), Lauren Brody, Lorin Sklamberg (also piano), Mattias Kauffman, Sy Kushner