Your dilemma: Your contact sent you a 72dpi GIF of their logo -- and it's spinning. Fantastic. Unfortunately, you're working on a print piece and what you really need is a vector file. You ask for the EPS but they don't know what that is and claim one doesn't exist.
Mmhmm.
So, how do you obtain the logo file you need? There are various corporate logo databases online. Most suck. They are a collection of half-ass attempts at recreating icons and typefaces. Search them first if you like, but chances are you'll be unhappy with what you find -- if you find anything at all. Instead, turn to your client's website. No, don't drag a logo off the homepage. That's how you ended up with the atrocity in your inbox. Instead, search the site for some PDF files. See where this is going?
Find a PDF with the logo visible. Open it in your browser, Reader, or whatever PDF viewer you prefer. Zoom in and out on the logo itself. If it looks good at 25% and 1,000% it's most likely vector art. If it gets pixelated it was rasterized before the PDF was generated and won't serve your needs. Find a good one? Good. Let's get that logo out of the PDF.
You can typically open PDFs in Illustrator or other vector-editing applications. I'll assume you have Illustrator. If you don't, play around with whatever you do have -- but you're on your own. In Illustrator attempt to select the logo. If the entire contents of the page are selected start breaking things apart. No, not your physical surroundings, but the visual elements of the file. Use the "Release Clipping Path" and "Ungroup" option until you can select just the logo itself. As you break up the elements you may notice large selectable areas that appear to contain nothing. Just delete them so they're out of your way. But be careful not to delete portions of the logo in the process.
When you're able to select the logo alone, copy and paste it into a new file. Save it as an EPS. Done.
You may need to do some additional work to ensure the colors match PMS values defined in the organization's brand standards. But if your contact sent you a GIF to begin with they undoubtedly have little concern for color matching or quality. So, you're probably in the clear.
Note: You can also open PDFs in Photoshop and other photo-editing applications. If that's your only option and your application can open PDFs you won't get the EPS, but you can probably get something better than your GIF alternative.