Very easy to find bottled water, in fact there will be some in your fridge in your cabin. Honestly the tap water was just fine, so consider that as well.
Checking in I had two large bottles of water with us. Security said nothing. They don't care about water just can't bring liquor. Our first day in port we bought a 12 pack of bottle water and took back to our cabin. I think I spent a whole $2.50 for a 12 pack. It was nice to have bottles to take with us each day we were in port. Yes it is also available on the ship.