Is there anywhere to get hot water in the parks?

Most of the locations have a handle on their sinks for hot water. They use hot water to clean off at the end of the day. (friend works in Disney and advised me). You can also find a tea/coffee vendor and ask them.

Absolutely!

Money talks

Ask where they have coffee or tea.

Yes! Check With The Different Vendors More than Happy to Help

Yes but you need to be creative about it. Only the baby center gives you hot water for obvious reasons. But you can but yes bags and use the hot water from that if you want?

Ask at any counter service or sit down table restaurant. Look for places that sell hot coffee and tea.

Yes

Yes in starbucks

There are also Starbucks and Joffreys Coffee places inside/right outside the parks.

I'm sure 1 of the restaurants could give you hot water

You could ask the restaurants