Yes, "on" is within the verb phrase. "Put" takes two complements, a direct object and a directional locative. In "put item on hold", "item" is the direct object and "on hold" is the locative. "On hold" is not an adverbial, but rather a verb complement. It is not clear that it has the sense of a locative here, since it is part of an idiom.
"Item on hold", the two verb complements, are what McCawley calls remnants of a diseased clause, the meaning of which is that the item goes on hold (status). That clause can be modified by a durational time adverb, like "4 days", for instance:
They put the item on hold for 4 days.
There is a potential ambiguity here, with the less likely interpretation being that the action of putting the item on hold was some sort of repetitive action, so that "for 4 days" is logically a modifier of the verb phrase "put the item on hold". But the more likely interpretation is that "for 4 days" modifies the verb phrase of the understood clause "the item [went] on hold", so that the item kept its hold status for 4 days.
Another fact that supports "on hold" being a complement rather than an adverbial is that it is present obligatorily:
*They put the item.