I'm not sure exactly. They are just like regular retail stores, and so probably open no earlier than nine but no later than 10. I would just call any of the stores in DD and ask them.
I would think 9 or 10 as well. But the Rainforest Cafe is open early for breakfast, so their gift shop would be open when they open (around 7am). Just in case you are looking for somewhere to hang out before the other shops open ;-)