If you're staying on property and have a dining plan, your "FREE" refillable mug on the way out the door everyday is a good start. But I'm fairly certain, that no matter which park you're in the price is the same, like a bottle of water or soda. Starbucks will let you use a snack credit for ANY size coffee. Hot or cold. I get a trenta cold brew light ice. It's massive full of caffeine and really really good. Far better than average iced coffee.
More free-er, byo to the resort and utilize the coffee maker in your room each day! Bring some inexpensive paper cups and lids from home, if you don't want to carry your plastic cup all day.