Virtually every think tank has a need for interns.  Below is a select list of some of the higher profile groups.  Review their websites for summer internship details starting in January.  You may need to poke around the website to find the internship link. They will likely want a short writing sample to demonstrate your writing and research skills.

Aspen Institute
American Enterprise Institute or
Brookings Institution
Carnegie Endowment for International Peace
Cato Institute
Center for Strategic and International Studies
Center for American Progress
The Clinton Foundation,
Henry L Stimson Center
Heritage Foundation Internship
Hoover Institution On their webpage see “Jobs at Hoover”
Hudson Institute
Institute for Policy Studies
Joint Center for Politics and Economic
Institute for Women’s Policy Research
New American Foundation
Progressive Policy Institute Review
Woodrow Wilson International See “Research Assistants” and also “Program Staff
Center for Scholars Internships” for specific programmatic opportunities
The Urban Institute Review
National Conference of State Legislatures-
National League of Cities
National Association of Counties
National Governor's Association
American Legislative Exchange Council
The Manhattan Institute for Policy Research
The Bill of Rights Institute

The Federalist Society

The Jack Miller Center

Pacific Research Institute

Ayn Rand Institute

Heartland Institute

Want more?  The Think Tank and Civil Society Program at the University of Pennsylvania has a Ranking System for Think Tanks