Last I heard, Allen Bradley had +40% of the US market. that was about 10 years ago, but that figure had held for a long time.
One way to tell who has the largest installed base is to look at the number of after market products to go into them. For instance, AB has an entire second-tier vendors association with periodic conventions etc. Omron, although big elsewhere in the world, has no such thing in the US.
"Venditori de oleum-vipera non vigere excordis populi"