Learn Russian: Мы ищем именно эту улику.

While both can be translated as exactly, именно is used to single out a specific item from a group (specifically this clue, not another one). Как раз is used more for coincidences, exact timing, or physical fit (such as clothes being just the right size), so it wouldn't make sense to use it to point out a piece of evidence.

Does именно have to go right before эту улику?

Yes, specification particles like именно must immediately precede the exact word or phrase they emphasize. If you moved it before the verb (Мы именно ищем эту улику), it would change the emphasis to the action: We are specifically LOOKING FOR this clue (as opposed to hiding it or destroying it).

Why is the noun улика in the accusative case (улику) here? Doesn't the verb искать sometimes take the genitive case?

You use the accusative case here because they are looking for a specific, tangible object. The particle именно heavily highlights this specificity. You generally only use the genitive case with искать when looking for something abstract, uncountable, or indefinite, like seeking truth, peace, or any good doctor.

What kind of clue is an улика? Can I use it for a puzzle?