Could Congress pass a law that allowed the Territorial delegates that are already elected to vote and have all the normal privileges other representatives have, or would a constitutional amendment be needed?
Yes and no. For final passage, yes an amendment would be needed. However, delegates can have all the other privileges including voting in committees, including the Committee of the Whole House which handles most House business. (The Republicans traditionally don't allow delegates to vote in the Committee of the Whole while the Democrats give them a limited vote that immediately gets redone without them if they provided a deciding margin.)