Yes, what the moving company will do is put your stuff in their storage facilities until you have a place to live, and then deliver them. Larger moving companies generally have storage facilities around the country, or have agreements with storage facilities.

However, be aware that when you find a new place to live, the moving company might not be able to deliver your boxes exactly when you need them. They will need to schedule your delivery around their other moves. So when you’re looking at this sort of arrangement with moving companies, be sure to talk to them about when to expect your belongings after you notify them that you’re ready to receive them.

Also, be aware that you will pay a bit more for this option, because you will pay for the moving company to pick up your items in storage and deliver them to you. It's like two moves: the pickup at your residence and delivery to storage, then the charges for cross country moving. 

A possibly more economical alternative is mobile storage options like PODs, particularly for smaller moving like yours. These companies will delivery a storage facility to your home, you fill it, they pick it up and store it, and then deliver it to you when you need your items.

Even if you don't like the idea of loading the storage container yourself, you can hire temporary moving help to do it for you.