They introduced a new model, the Family Rotary, or FR, which was a very well designed and strong machine, in the late 1890's and manufactured this design, with some variations, up until WW II.
Unfortunately, there is nowhere near as much information on the White Company as on Singer.
White was, next to Singer, perhaps the largest and best known of the US sewing machine companies.
Unlike Singer, they did not offer many different models, or change models very often.
White owned their own forests and operated their own cabinet factories, rather than contracting this out.
These shuttles and bobbins are not available and somewhat difficult to find. I always tell people that "White is Singer spelled backwards...".
Should you locate and consider a White vibrating shuttle machine, be sure it has it's shuttle and a bobbin or two. Everything about the White is the reverse of the Singer...
White took over the Domestic and King sewing machine companies and eventually became White Consolidated Industries. There is one serious consideration as to the VS machines... The shuttle has a post in it, and the bobbin has a hollow core.They ceased manufacturing in the United States sometime after WWII, probably the late 60's or early 70's. The bobbin is loaded into the shuttle by sliding onto the post.