tl;dr: A lot of soy ink is used, but mixed with petroleum-based ink.
According to Wikipedia:
A major problem with soy ink is that it takes more time to dry than
petroleum-based inks, due to its lack of evaporative solvents in the
form of VOCs.
"VOCs" are Volatile Organic Compounds -- organic in the chemistry sense, not the farming-process sense.
This drying problem means that the use of soy ink is limited to purpose-made printers and/or papers.
While VOCs can be both synthetic or natural, the synthetic variety are associated with long-term health risks, air quality problems, and carbon emissions. So adding synthetic VOCs to soy-based inks to eliminate the need for special equipment or papers would defeat the purpose.
One way around this problem is to simply produce soy ink that's only partially soy ink. The standard for what percentage of an ink must come from soybeans is set by the American Soybean Association (ASA), which authorizes the use of this seal by inks which meet the standard:
In the standard they publish, no ink needs to be more than 40% soy to earn the seal, and some specialized inks (such as heat-set ink) can earn the seal even with 10% soy or less.
In fact, it seems that this mixing is done across the industry. According to the same Wikipedia entry, soy ink was developed in the 1970's to hedge against rising petroleum costs. Use of soy ink (mixed with conventional ink) is widespread in the printing industry -- according to this article from 1999, at the time 90% of daily newspapers and 25% of commercial printers were using soy ink, purchased from an industry with over 100 producers.