Getting permission to use copyrighted works is a simple two-step process, in theory. You find who owns the copyright and you contact them to ask.
For mainstream publications the copyright owner is almost always either the author or the publisher, and the publisher is the simplest place to start as they can refer you, if needed. Many publishing companies have offices for copyright/permissions with contact information you can find on their website.
For cases where the copyright owner/publisher is hard to identify, things can get complicated. There is no one way to sort this out. It depends on what clues you have from the work in hand. Basically, it becomes a little research project of its own. The Library can help with this. Also, the U.S. Copyright Office will research copyright ownership for a fee.
Once you have found someone to ask, you will find that response time varies considerably. Anything from a few hours to a few months is certainly possible.
Be aware that inability to identify the copyright owner does not excuse you – legally speaking – from getting permission; nor does silence imply permission.
Sample Permissions Letter
Dear Sir or Madam:
I am writing to request permission to copy the following for use in my classes next semester:
Published in: The American Journal of Economics and Sociology, vol. 57, no. 1, pp.71-94
Copyright: The American Journal of Economics and Sociology, Inc., 1998
Distribution: The material will be distributed to the students in my class from within the password-protected learning management system.
Type of reprint: digital copy or scan
Use: Supplementary teaching materials
I hope you will consider this request. Your response by ___[date]___ would be appreciated.