Chromebooks have come from nowhere to grab nearly a fifth of U.S. school purchases of mobile computers, posing problems for Microsoft Corp. and possibly even Apple Inc.

The inexpensive laptops, which run Google Inc. software but are mostly sold by other companies, accounted for 19% of the K-12 market for mobile computers in the U.S. in 2013, according to a preliminary estimate by Futuresource Consulting. In 2012, Chromebooks represented less than 1% of the market, according to the research firm, whose estimate includes both...