In many cases, requesting permission to use published materials is straightforward. It’s best to start with the article or book publishers’ websites, which often have Web forms for making such requests. For example, here is IEEE's page spelling out how to make publication re-use requests.
In other cases, you might need to do some digging first to identify the copyright holder. Below are some suggestions if you need help. For more background, you may also wish to check out Peter Hirtle's Copyright & Cultural Institutions, Chapter 8.
U.S. Copyright Office Database: Search a public database of all works registered with the U.S. Copyright Office after Jan. 1, 1978.
Materials from an Archive or Library Special Collections: It is possible (though not guaranteed) that the library or archive you used has donor information (if the donor was also the copyright holder), or transfer data that accompanied the acquisition of the collection.
WATCH File: The Writers, Artists, and Their Copyright Holders database contains primarily the names and addresses of copyright holders or contact persons for authors and artists whose archives are housed, in whole or in part, in libraries and archives in North America and the U.K.
Copyright Clearance Center: Commercial services like the Copyright Clearance Center can, for a fee, streamline the process by searching for the rights holder and securing the license for you.