How about we limit this thread to just companies that do business directly with the general public?
Demoulas/Market Basket is a ever-expanding New England grocery chain, with a gross over $3 billion, number 125 on the Forbes list of private American companies, and as this wiki shows, no website at all.
I’ve shopped there for most of my life, and although their stores are fairly modernized, they run a tight ship with few bells and whistles. No user loyalty cards, just low prices. I think the lack of a website is part of their frugal management style.