I'm looking for data on how far individuals will travel to reach a bus
stop or rail station in the USA.

This is called the "catchment area" of the station.  The answer
depends on the "mode" of travel (e.g. walk, bike, drive, connecting
bus, etc).   With Google you'll find many non-specific references and
generalities relating to this topic (for example "people will walk
about 1/4 mile" or "bicycles help extend the catchment area of a
commuter rail station").  But I'm looking for some hard data, based on
surveys.

Here is a typical "theory based" treatment of the topic:
http://www.fhwa.dot.gov/planning/toolbox/orange_application_walking.htm
http://www.fhwa.dot.gov/planning/toolbox/orange_figure8b.htm

And here's a reasonable walking survey from Canada:
http://www.enhancements.org/trb%5C1538-003.pdf
I did not find anything useful at: http://trb.org/