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Yes Minors can.

“The files may contain additional information including “original copies of the associated documents,” the report added. Such documents include birth certificates, naturalization certificates, or oaths of allegiance for U.S.-born persons who adopted the citizenship of a foreign country as minors. “

To give further context along with Obama’s OWN ADMISSIONS:
“The passport files include “personally identifiable information such as the applicant’s name, gender, social security number, date and place of birth, and passport number,” according to the inspector general report.

The files may contain additional information including “original copies of the associated documents,” the report added. Such documents include birth certificates, naturalization certificates, or oaths of allegiance for U.S.-born persons who adopted the citizenship of a foreign country as minors.

The Obama campaign acknowledged at its “Fight the Smears” Web site that Obama was a foreign national until the age of 18, by virtue of his father’s British then Kenyan citizenship.

“Since Sen. Obama has neither renounced his U.S. citizenship nor sworn an oath of allegiance to Kenya, his Kenyan citizenship automatically expired on Aug. 4,1982,” the Web site stated.

“Fight the Smears” attempted to debunk rumors that Obama was not a U.S. citizen by producing a 2007 computer-generated copy of his certification of live birth. “

All from: http://www.newsmax.com/timmerman/brennan_passport_breach/2009/01/12/170430.html

I misspoke and meant to say that it couldn't be done on their behalf. However, as below will show, it may be practically impossible for a minor to do so, especially a young child.

F. RENUNCIATION FOR MINOR CHILDREN Parents cannot renounce U.S. citizenship on behalf of their minor children. Before an oath of renunciation will be administered under Section 349(a)(5) of the INA, a person under the age of eighteen must convince a U.S. diplomatic or consular officer that he/she fully understands the nature and consequences of the oath of renunciation, is not subject to duress or undue influence, and is voluntarily seeking to renounce his/her U.S. citizenship.

http://travel.state.gov/law/citizenship/citizenship_776.html