I think it should always be “contact me” as contact is a transitive verb and should be followed by an object. But could someone clarifies? Is “contact with me” correct usage in some context?
When used as a verb, “contact” is always transitive. As such, you are correct: it should always be “contact me.”
Note that the phrase “contact with me” may appear as an object in other situations, where it may be mistaken for a verb. Example: The other day, my long lost brother made contact with me. In this example, made is the verb, and contact is its direct object.