I am a US citizen who lives in the US.  I used a credit card to
purchase a computer data projector in Japan today for the yen
equivalent of about $2000 US.  I paid about $100 tax.  I have the
receipt.  I then ran to catch my plane back to the United States and
did not have time to apply for a VAT refund at the airport.  Is there
any way for me to get a VAT refund from California?

I am pretty sure that you are out of luck.  The same situation
happened to me when in Canada and you have to so all the paperwork and
submit it before you leave the country.  You can't file it from USA. 
Now I don't know if you could file it if you returned to Japan within
a reasonable amount of time, say a month or less.  I would be very
interested to find out what other people know.  I did buy a camera
flash in Osaka for about Y30,000 back in the year 2001 before my wife
and I got married, and I don't remember how much VAT there was if any.
 But I didn't even think about applying to get a refund.

weegee