YES
A Lego museum is actually in the works at the Smithsonian. There are other unofficial museums, there is one in Chicago and another one in Ohio. The Guggenheim Store has a Lego museum exhibit.
No there is not a mob museum in Chicago
Chicago Children's Museum was created in 1982.
Museum Campus Chicago was created in 1998.
The web address of the West Chicago City Museum is: www.westchicago.org/museum
The web address of the Chicago Childrens Museum is: www.chichildrensmuseum.org
Chicago Postcard Museum was created in 2007-11.
Chicago History Museum was created in 1856-04.
Chicago Museum of Science The Museum of Science and Industry
The exhibitions at the Chicago History Museum are constantly changing. Currently the Chicago History Museum is featuring: Inspiring Beauty, 50 years of Ebony; Shalom Chicago, Vivian Maier's Chicago, and Lincoln's Chicago.
The web address of the Chicago Children'S Museum is: http://www.chicagochildrensmuseum.org
The web address of the Chicago Sports Museum is: http://chicagosportsmuseum.net