I once contacted Dave Budeau, the manager of EE Wilson WA about hosting private events at EE Wilson. I did not mention anything about airsoft at the time and decided not to pursue it further. Here's what he sent me:
To answer your questions. Generally, hunting can occur from Sept-Feb, with October the busiest by far. From March thru mid-August, we encourage visitors to stay on area roads and improved trails to minimize disturbance to young and nesting wildlife.
Dave Budeau, Mgr
E. E. Wilson WA
Contact email at the time was
Eewilsonwa@aol.com.
That being said, EE Wilson is very open and flat, making it less than optimal for airsoft use. The lake area sees significant public use all year and many areas of the wildlife area are thick with poison oak.
Playfields on private land are better for everyone. Public lands may be viable, but should be more remote and discreet than this location.