Is with a shovel a noun phrase?

Yes, 'with a shovel' is a noun phrase. The noun 'shovel' is the object of the preposition 'with'.

A noun phrase is any Word or group of Words based on a noun or pronoun that can function in a sentence as a subject or an object. A noun phrase can be one Word or many Words; it can be very simple or very complex.

• She is here.
• Where is the copier?
• We can dig these rocks with a shovel.
• That car with the ticket on the windshield is mine.