We recently purchased the company in 2013 and cannot find much on our history, and since we know redditors seem to find it all if they don't already know it, we're seeking your help.
Here's what we know:
Your best bet would be down at the NYC municipal archives, I think.
"The former bootblack was Anthony Aste, 88, founder of the Griffin Manufacturing Co. (the world's largest makers of shoe polish)"
Anthony Aste's Patent for a Shoe Polish Stand
6 Patents registered to Anthony Aste, all shoe related
Article from Brooklyn NY Daily Eagle Jan 25, 1955, Robert L. Aste's succession to replace his father as president of Griffin.
Article from Brooklyn NY Daily Eagle March 31, 1996, regarding how the Two-in-One Shinola Bixby Corporation stole a formula for shoe polish from Griffin. Description of court ruling.
Trademark case surrounding Griffin's attempts to protect the use of the terms Allwite and All White.
Cincinnati Radio Station WLW-AM notes that:
1941: Singer Andy Williams and his three brothers came to WLW-AM in 1941, after Chicago’s WLS-AM canceled their radio show. They sang on the 15-minute show called “Time to Shine,” sponsored by Griffin (shoe) Polish at 8 a.m. before going to school.
Fifty years later, Williams sang the opening theme song for WVXU-FM’s “Cincinnati Radio: The Nation’s Station (1921-41)” documentary: “It’s time to shine! So shine your shoes, and you’ll wear a smile. Shine your shoes, and you’ll be in style. The Sun shines East,and the Sun shines West … Griffin Polish shines the best!”
An old ad you may have seen before for Griffin Microshine.
Griffin Shoe Polish Boy Scout fundraiser from 1975.
Hope these help!
You might try the Library of Congress' newspaper digitization project. You'll only get results through 1922, but you can limit the search results by U.S. state. I see a few references to a Griffin-White Shoe Company, but without a little more information on the history of the company name it's hard to tell if that's the same one.
Edit: I don't have much time to spend on this, but I just took a look at the results doing a search for "Anthony Aste" or "Tony Aste," and you might find those searches more useful. Here's an article with a brief biography calling him "King of the Bootblacks."