"Search this website" or "search in this website"

Sometimes I see the following expression after a search box:

Search this website.

Some people say that the correct expression is

Search in this website.

Which is correct, and what is the grammatical explanation?

Search can be either a transitive verb or an intransitive verb, that is, it can have a direct object or not. A direct object is the thing to which the verb's action is directed. So,

Search this website

is a valid use of the transitive search, since the searching is directed against the website.

Search in this website

is also valid, grammatically. This is essentially using search in an intransitive way, but then attaching the prepositional phrase in this website to add information, namely where the search is to take place. Since where the search takes place is the same thing as what is to be searched, the two forms have the same meaning. The simpler, transitive form (search this website) is generally to be preferred.