From another site:
There are numerous shipping companies who ship to Puerto Rico and many options. Full service moving companies are the most expensive. These companies show up at your home, professionally pack your goods, wrap your furniture and load it onto a truck. Your belongings will be driven to port, placed on pallets, wrapped, weighed, and then loaded into a shipping container. Expect to pay $2,000 and up for this service, depending on the amount you are shipping and your distance from port. Check your local listings for movers who will transport your belongings to the closest port and ship them.
Moving Companies
Other options include a container drop-off. This is more affordable. The container is dropped off at your home and you load it yourself. You can chose a 20-40 ft. container, which is enough for a small to medium sized home. Pick it up and ship it for you. When it arrives in Puerto Rico, it is dropped off again for you to unload.