The American Typewriter comes by default on Mac machines, but not on Windows. If those in my agency who need to use this font on their PC, do they have to purchase a license for it, or could I just give them the font to use?
In terms of technology, likely no problem. Font file formats are mostly cross platform these days (OSX and Windows 7 can handle most font file formats). Legally, though, it's murky. The fonts that come with OSX are licensed with the purchase of the Operating system. Technically, you are allowed to use them with the operating system...but not on other machines.
You can't just give them the font to use. The OS X EULA doesn't allow that, so it would not be legal, but there is the practical point that the older TTC (TrueType Collection) format, which is how this particular font is store on the Mac, is both rare (although it's used on both platforms) and obsolescent.
There are so many advantages to OpenType, beside the simple fact of it being a truly system-agnostic wrapper for both Type 1 (developed by Adobe) and TrueType (developed by Apple and Microsoft) outlines, that I advise designers using any font in a production environment to upgrade. In your case, you would be best to use the OTF version in both environments so that you don't run into reflow issues moving documents from one platform to another. If you are likely to be creating documents in more languages than just English, or even if you have occasional need to set foreign words, you have even more reason to opt for OpenType.
Truetype or Postscript fonts are copyrighted works. You are not allowed to copy them without a license, and that obviously includes copying them to multiple PCs. The license coming with MacOS X actually says that you are not allowed to copy them, but even if it didn't specifically say that, as long as you don't find anything in writing that allows it, copying fonts is illegal.
The default Mac American Typewriter is a TrueType font. In most cases, TrueType and Postscript fonts are not compatible between the Mac OS and Windows. So the Mac TrueType font generally will not work on a Windows machine.
[Note: the Mac can use either a Mac version or a Windows version of almost any font. Windows is not that forgiving and can not use Mac versions of fonts.]
You'd be better off purchasing an OpenType version of the font for the number of seats you need. From here for example
The OpenType format will work on both the Mac and the PC and will be constructed much better than the default TrueType format which ships with the Mac OS.