An Israel-based supplier of autopilot components, Mobileye NV, has secured agreements with two auto makers to provide systems for fully autonomous cars in 2019, deals that give a clearer timetable for when auto makers believe they can start producing vehicles entirely capable of driving themselves.

While there are many Silicon Valley companies, including Google parent Alphabet Inc., racing to develop autonomous-vehicle technology, Mobileye is considered a leader given its current share in the market for components needed to...