When taking a picture with old fashioned film what sets the resolution of the picture? Is it the wavelength, or the chemical makeup of the film?
Both the diffraction limit and the grain size could affect the resolution limit of analog pictures. Let's see how they compare.
Diffraction limit The Abbe diffraction limit states that the size of the spot is $d=\lambda f\#$. The $f$-number will depend widely for different cameras and illumination settings. If, for example, you use $f/10$, this gives about 5μm for green light. See Edgar Bonet's comment below for more details on different lenses.
In short, both the diffraction circle and the grain size vary from a fraction of a micron to a few microns. The resolution will be determined by the both of them.
You can also check out http://en.wikipedia.org/wiki/Science_of_photography. There's some further info on this topic.
EDITS: I made several edits thanks to the commenters.