I want to know if and where I can subscribe to a statistics service
that will provide either real-time or aggregate statistics on
professional football games (read: NFL) for the upcoming season.

This service should be accessible programmatically, via a web service
or similar data feed that I can subscribe to and use to create my own
web site (or whatever.)  The data should include statistics for each
player involved in each games and would include things like "number of
carries" "total rushing yards" "number of receptions" "number of
touchdown passes thrown" and so on.

My guess is that large fantasy sports organizations (e.g. Yahoo!) must
have access to a service of this type, and furthermore, I have to
imagine the NFL provides a service like this, but I don't know for
sure.  If there is an official NFL web service, that would be my
preference, but if other providers exist, I would like to know about
them as well.

There is a company called stats, inc. (www.stats.com) that can provide
you with this service.  From their website they are the stats engine
behind heavy hitters like yahoo.com, espn.com, and more.  Following
the link below will take you to the business portion of their site
where they talk about the applications that you are looking for.

http://biz.stats.com/internet.asp

As for the nfl.com site it is actually powered by cbssportsline.com. 
So no luck in getting the stats straight from the NFL.

I hope this helps!