The simple answer is no.
The definition of "plugin" is well understood as a component that provides additional functionality to a page through its actions. It’s wise to expose the availability of individual plugins to the user interface so that it can make informed decisions about the types of content displayed.
"Installed search engine" is not defined and differs in different browsers, and also does not matter for the functionality of a single page or site; this is a user parameter that you do not expect the page to know about, just like (for example), you cannot get a list of current printers.
You should be able to write a plugin that JavaScript can use to access this information (which will probably require access to registry keys for each browser installed), but this will rely on the user installing it in the first place.