Whitman's writings are out of copyright. So, yes, you can quote his words without seeking permission from anyone.
Editorial introductions, annotations, the biography, and the criticism found on this site are under copyright. You can certainly quote from these items within the "fair use" limits of copyright law. If you wish to make more extensive use of these materials, contact the directors, Matt Cohen <matt.cohen@unl.edu>, Ed Folsom <Ed-Folsom@uiowa.edu>, or Kenneth M. Price <kprice2@unl.edu>.
In general, for permission to use photographs you should write directly to the library holding the photograph (libraries are credited within the annotations in the gallery section of the Archive). Any photos listed from the Library of Congress, from the National Archives, from the Gay Wilson Allen Collection, or from the Ed Folsom Collection can be reproduced without permission.
The Whitman recording of “America” is, we assume, in the public domain. Since its rediscovery in 1992, it has been used by singers on records, by compilers of CDs of poets reading their works, and even by advertising firms (it was used in a popular 2009 Levi’s commercial). No one has claimed ownership of the now-lost original wax cylinder, which was first played publicly as part of an NBC radio program in the early 1950s.
Most contemporary criticism is copyrighted by the author or by the publisher.
We regret to say that we have neither the resources nor the time to help you with your research.