If you determine that your use is not fair, or if you think it is fair but you'd like play it safe, you'll need to ask permission from the copyright holder. Columbia University has excellent information about getting permissions including example permissions letters.
Step 1: Identify the copyright holder
Believe it or not, this can be one of the most difficult parts of the process and it is the reason you'll want to start pursuing permissions as soon as possible. The copyright holder might be the original artist or author, their heirs, or, most frequently, their publisher. If you need help figuring out where to start, please get in touch and we'll figure it out together!
Step 2: Ask for permission
Generally, you'll want to write a nice letter to the copyright holder explaining what you want to use and how you want to use it. Make sure you provide as much detail as possible -- if you plan to reproduce a color image in black-and-white, let them know that, for example. Make sure you let them know that you're asking to reproduce something in an unpublished undergraduate thesis.
Step 3: Negotiate (if necessary)
Depending on the copyright holder, the first response you get from them may ask for payment in order to grant the license to reproduce the work. You should absolutely feel free to contact them again, perhaps by phone if they provide a number, and explain very clearly that this is for an unpublished undergraduate paper.
Step 4: Repeat as needed
Sometimes you'll hear back from the person or company that they no longer hold the copyright. That puts you back at Step 1.
Document your search!
Keep copies of all your correspondence.