Ice Tea is included, but only at meal times.
A refillable mug wouldn't help with getting ice tea, it's for soda pop from a fountain.
There's a 24 hour drink station on deck 9 aft and they have coffee (deacf and reg), hot water, ice, cold filtered water, and they have several types of tea bags (decaf and reg) and they have packets of instant hot chocolate.
You could make your own tea with the hot water and a tea bag and then pour it over ice, if that would help.