An adverb of place is usually located at the end of a sentence.
'Everywhere' is an adverb of place (adverbs that end in -where refer to a location without specifying a specific location: somewhere, everywhere, nowhere, anywhere, etc.,).
In the given sentence, 'everywhere' is placed in between a prepositional phrase ("for the keys" is not the object of the verb "looked", it is a prepositional phrase that complements the verb).
Hence the sentence requires change and option D is incorrect.
The adverb is placed at the end of the sentence in option B, which is right; an adverb of place is generally written at the end of a sentence. Hence B is the correct option.
An adverb of place is not located before the verb, but after. Hence option A is incorrect.
In option C, the adverb is placed in between the prepositional phrase "for the keys", it seems to modify either the preposition "for" or the object (the keys) of the preposition. Adverbs modify neither prepositions nor nouns. Hence option C is incorrect.