I search somewhere for something I want to find. So I search my house for my keys. What about a query I type in the search bar?
I search "what is ve day" on Google
I search for "what is ve day" on Google
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Personally, whenever I explain a search query verbatim to someone, I don't use "for":
I searched "How old is Leonardo DiCaprio" on Google...
Essentially, I'm treating the full search like direct speech.
When one merely states the topic/person/object that they searched for, I feel like without the word "for," the sentence doesn't seem quite right.
I searched for cat images on Google.
I searched for Leonardo Dicaprio's age on Google.
Maybe this is governed by personal preference; I'm not too sure.