Le he estado buscando por todas partes.

Why not this: Yo lo buscaba por todas partes. (imperfect tense)?

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Imperfect means it happened in the past only. As written, the action is continuing into the present.

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Why le instead of lo?

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le is often used instead of lo; it’s called leísmo.

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English Original Sentence

I’ve been looking for it everywhere.

In fact, using le (him) for an inanimate object is not proper Spanish; it should be lo. I can say that even though I’m not a native speaker.