The short answer: it depends. While yes, many movers can go to a storage unit, not all will, and it typically comes down to the company and service type you choose. Movers are no strangers to loading into or out of storage units since so many people rely on them. However, not all moving companies can or will accommodate this request, and others may charge extra.
If you need movers who go to storage units during your move, confirm the availability of this service with companies well in advance. You can usually give the moving company a call and ask if their movers will go to a storage unit if this information isn’t already on their profile or company pages.