Military Voting Support Act of 2001

Amends the Soldiers' and Sailors' Civil Relief Act of 1940, for purposes of voting for any State or Federal office, to prohibit a person in military service absent from a State because of compliance with military or naval orders, from being deemed, solely by reason of that absence, to have: (1) lost a residence or domicile in any State; (2) acquired a residence or domicile in any other State; or (3) become a resident in or a resident of any other State.

Amends the Uniformed and Overseas Absentee Voting Act to require each State to: (1) permit absent uniformed services voters to use absentee registration procedures and vote by absentee ballot in general, special, primary, and runoff elections for State and local office; and (2) accept and process, with respect to any such election, any otherwise valid voter registration application from an absent uniformed services voter, if the application is received by the appropriate State election official before the election.

Amends Federal armed forces law to authorize: (1) the Secretary of a military department to make a building located on a military installation under the jurisdiction of the Secretary available for use as a polling place in any Federal, State, or local election for public office; and (2) the Secretary of Defense to make reserve component facilities available for the same purpose.

Requires each State, with respect to any recently separated uniformed services voter requesting to vote in a Federal election, to: (1) deem the voter to be a resident of the State; (2) waive any residency or domicile requirement; (3) accept and process any otherwise valid voter registration application from the voter on the day of the election; and (4) permit the voter to vote.

Directs the Secretary, during Federal elections in 2002, to establish a demonstration project to determine the feasibility and advisability of using an electronic voting system by uniformed services voters during such elections in 2004.