No, the sensors in mirrorless cameras are not exposed to the sun at all times! You don't have to worry your film getting exposed without a lens cap, when shooting with a mirrorless film camera either!
Though if you camera has a cloth shutter, the sun can burn pinholes or worse in the shutter. I had it happen to a Canon rangefinder recently. Current mirrorless cameras do not use cloth shutters so there is far less chance of such a problem.