The Program refers to the GPLed code, which they don't necessarily need to modify in order to make AP software. If so, they can just send you a tgz of a very useless set of already-available unmodified GPLed source.
The "Program" refers to the binary that is distributed. In the case of a binary linked against GPL code, it is considered to be a derivative work. LGPL makes an exception that covers dynamic linking only.