According to the latest membership list, Chelsea Clinton is a member of the Council on Foreign Relations (CFR). Her father, former President Bill Clinton, is also a current CFR member. [Last year the Democratic Leadership Council, or DLC, which was closely aligned with Bill Clinton, closed.]
Chelsea earned a Master of Philosophy in international relations from the University of Oxford in 2003, and she has some serious foreign relations "street cred" with her mom as Secretary of State.
Chelsea is a special correspondent for NBC News. Perhaps her good friends in the media who are members of CFR (there are dozens) convinced her to join the think tank. In addition, plenty of former Clinton Administration officials are CFR members.
Hillary Clinton is not a current CFR member, but she said back in 2009 that the State Department "gets lots of advice" from CFR.
As Secretary of State, Hillary Clinton is a Public Member of the Wilson Center. Also, you can read a previous Think Tank Watch post about Hillary Clinton's think tank here.