The store (and company) wants to cut down on the amount of plastics that they use. This is a positive thing and much better for the environment. This is not debatable.
Now, when one is shopping at the mall (most Disney Stores are in malls), odds are you have already purchased something at another store, and have it in a bag. When you buy a shirt, or mug, or whatever at the Disney Store, you don't need another bag. You can just put it in to the bag you already have. If The Disney Store didn't charge for the bags, people would mindlessly (out of habit, perhaps) accept the bag from Disney without thinking about just using the bag they already have in their hand. Additionally, if Disney only charged 10 cents or 15 cents, it wouldn't be enough for most people to even notice or care about, so they might just get the bag, even if they don't need it, out of habit or whatever. But, one dollar is enough to force people to think about whether they really need the bag or not, so I'm sure that is why they decided on that amount.
Without financial incentive, people will rarely change their bad habits.
This thread is proof of that.