Yes, of course you can. Sounds like a silly question, but we get asked this all the time. If you are paying to have something designed, why wouldn’t you be able to have the artwork? If there are designers out there withholding artwork from you that you’ve paid for, you need to stop using them immediately! If you want to take your artwork and change designers, that’s not on you, that’s on the designer. They should have done more to ensure you stay a satisfied customer.
Whatever we design for our customers becomes their property. We will send it through to you for your records, and we keep a copy on our servers as a backup should you ever need it. (Not saying you’re going to loose it, but, it happens!)