Unfortunately, I haven't been able to locate a very similar vegan version. The best I can do is Rina Shah's thong with rivets (pictured right). Not exactly a perfect match . . . at all. It's made from man-made leather and, clearly, is decorated with rhinestone-ish rivets. At $55, it's a bit less expensive than the Jack Rogers' sandals.
Also not-so-similar are a variety of summer sandals from Madden Girl. First, there is the "Adaline" sandal. It is entirely man-made and comes in the pictured turquoise, pink, yellow, brown and black. $49.90; available here via Zappos. So similar I'm not quite sure why it has a different name is the "Bayleee" sandal. Also entirely man-made; it comes in turquoise, black, and yellow. Also $49.90; here. Third, Madden Girl makes a "Dandye" flip-flop faux leather sandal that is further away from the Jack Rogers' sandal, yet still seems tangentially similar in my mind. Also available in a variety of colors; also $49.90; available here.
Finally, the only way these Madden Girl "Bajjaa" sandals are like the Jack Rogers' sandal is that they are vaguely Native American themed, but I thought I would include them. They are available in the pictured black (which I like most because of the purple accents), brown, and white. Also vegan (obviously); also $49.90; also available through Zappos -- here.