Rockwell Collins Inc. is making a big bet on connecting aircraft to the internet at 35,000 feet with its deal to pay $6.4 billion for B/E Aerospace Inc., the world’s biggest maker of plane seats.

The deal would lift Rockwell into the elite tier of suppliers to Airbus Group SE and Boeing Co. as the plane makers boost production on a $1.5 trillion backlog of new jets built up during a seven-year “super cycle”.

Buying...