I am trying to add some imagery to my iOS/Android app and I like this kind of art:
Does anyone know what terms I should search for to pull up this kind of art? I want to search on shutterstock and similar sites.
I'd say it's merely a Monochrome Vector Illustration. Maybe slip in "cartoon" or "comic" as a keyword.
I don't think there's a specific name for the style itself.
You get much more to select if you take coloring book style graphics (all drawn with solid thin black strokes on a white background). Then you fill the areas with proper colors. Finally you remove the black strokes or recolor them to fit.
This all is technically trivial. You must probably close manually plenty of gaps to prevent the color to spread too widely, if you make floodfills with the paint bucket.
There are plenty of software for the job. It's possible to do it as well in vector domain as using raster images. The latter needs high resolution images for good results. Screenshots probably are awfully too coarse.
One of your major problems is proper licensing of the images. You must get a license that allows modifications and distributing it as a part of software. Totally free public domain stuff is a no problem-case. Do not even think to distribute well known cartoon figures without paying premium price for the right to use them - no matter do you sell something or do you publish freeware. If you see some fine image in the web and there's no overprint that say "this is not free, you must pay!" it does not make it free.