What values can I use for _entity_access in a configuration entity I am developing?
According to the Drupal 8 documentation:


  _entity_access: In the case where an entity is part of a route, can check a certain access level before granting access (e.g.,
  _entity_access: 'node.view').


The part after the dot is the type of access you want to check on the entity. To find the full set of possible values, you have to look at the code of the entity (in your case, in \Drupal\search\Entity\SearchPage). The default possible values are view, update, delete, create. Entities implemented by SearchPage can also be disabled and enabled. (See search.routing.yml and SearchController.)

See also EntityAccessCheck::access().