Hi there, Darcey!
We've certainly come a long way from more archaic ways of communicating on the sea. Yes, a mobile hotspot device or phone can usually work onboard a Disney Cruise Line ship. On the high seas, your ship offers Cellular at Sea. This service works reasonably well with most devices when used for voice calls and Short Messaging Service (SMS, the "green bubble" type of text message). Light data service is supported too, though this service usually isn't as fast and responsive for data service as we are used to from land-based cell towers. When at or near ports, I've found that I can usually connect to regular cell service, and even tether my laptop or other devices to my phone's hotspot. However, that setup usually works best on open decks and verandahs, where my device has a clearer path to towers on land, rather than at the desk inside our stateroom.
Before traveling internationally and at sea, be sure to check with the service provider for your device or mobile phone. Some carriers may charge extra for connecting to Cellular at Sea or roaming to other similar international services. I've noticed that some plans place caps on the amount of high-speed data that can be used per day, restrict hotspot tethering, or impose other similar limitations when connected to other services and carriers. So I highly recommend reading the fine print published by your specific carrier. If the costs and terms make sense for your needs, and your devices are compatible, you'll usually be able to get connected reasonably well while on international waters.
As an engineer who works with data communications in my job, I'm impressed with the ways Disney Cruise Line and partner providers have overcome significant technical challenges to provide connectivity while at sea. Between Cellular at Sea and the Wireless Internet Access onboard, my personal choice and recommendation for data is to use the ship's Wi-Fi. It's usually a faster service, even with the basic Internet Package. And I like the convenience of using the same DCL-GUEST Wi-Fi that is required for taking advantage of all that the Disney Cruise Line Navigator app has to offer. You can purchase length-of-cruise Internet plans or single-day service.
Tony's Tip: Even in airplane mode, you might find that your mobile phone may still be able to send and receive certain types of text messages when connected to DCL-GUEST Wi-Fi without purchasing an Internet plan. While I do love the Disney-themed emojis available in the Navigator app's included onboard messaging service, my family also usually relies on iMessage to communicate with each other on the ship and with friends and family back home.
Darcey, I get it. Sometimes we need to keep lines of communication open while on vacation. But I hope you find time to disconnect as much as possible and enjoy all the amazing experiences your Disney cruise has to offer.
Have a magical cruise, Darcey!
~Tony