I am trying to find the name of the company, I believe to be located
in California, that literally sends researchers into the field to
count the number of new housing starts by building company (for
example, Toll Brothers) across the country.  I believe the data is
published monthly and has a short lag time.

I beleive there are two companies that could possible meet your needs.
 The Meyers Group or Construction Monitor.  Based on your description
Meyers sounds like the appropriate choice.  They report on new home
sales contracts by builder by county on a national basis. 
Construction Monitor reports on building permits by builder by zipcode
on a region by region basis.  They primarily cover the west.