According to the Brand Underground article:
"From that first batch of a few hundred shirts in 2003, Barking Irons had seen its orders climb to 12,000 a season."
"Barking Irons popped up in GQ, Elle Girl, Maxim and elsewhere. The main character (played by Adrian Grenier) on the HBO show “Entourage” wore a Barking Irons shirt, and this fact was reported in People magazine. Late last year, the brand went global: a friend of the brothers’ helped coordinate a miniature trade show in Tokyo, leading to their first sales to Japanese retailers and a full-page spread in a Japanese shopping magazine."
So they're doing all right for themselves

They're in a lot of hot stores and I imagine most of their sales are through that rather than the website.
I agree that flash is a terrible choice for a website, and that maybe they'd be doing even better without it, but they are proof that you can use it and still make mad bank
That said the article also highlights some poor business decisions (getting a bad deal from their agent, leasing an overly large space) and setbacks (orders actually declined in the last year to 10,000 and a delay from a supplier). So maybe the flash site is just another one for that list.
Anyway... like Jasonda said "They are screen printed, possibly with water-based inks." I'd add possibly with disperse dyes, and that it's more likely to be with one of those two methods rather than plastisol inks (but it could be plastisol).