Hi Lucy!
Welcome to the Disney Parks Moms Panel! I completely understand the need for coffee in the morning. My almost three-year-old now says "Want get coffee? Want get latte?" to me on an almost daily basis, and I can't tell if I'm doing a great job at parenting or am already failing.
While there aren't any coffee machines available to rent through your Disney Resort Hotel, I do have a few ideas. You can certainly bring your own machine from home, or you can buy and bring a cheap machine with the intention of leaving it in Florida. As a bonus, the machine's void in your suitcase can then be replaced with souvenirs. Don't forget all of the coffee supplies you'll need. (Coffee, filters, etc.) There is a small section of the Ink & Paint shop, which is the souvenir and sundry store at
Disney's Art of Animation Resort Hotel, that has simple groceries you may need during your stay, including coffee, filters and creamer. However, to ensure that you have your favorite brew it's best to bring it with you.
You can also purchase a machine and supplies from an online retailer and have it sent to your Disney Resort Hotel prior to your arrival.
There is a $5 per-package handling fee, but this is a great option if you're not renting a vehicle to get to a local retailer. Pro Tip: look to retailers like Amazon or big box stores like Target who have free shipping after a certain amount spent.
You could always hire a taxi to take you to a nearby store. There is a SuperTarget just west of Disney's Animal Kingdom that will have everything you need, plus some things you didn't. And as a last resort, you could always request a Little Mermaid room closest to the lobby to make the walk that much shorter.
I hope this helped, Lucy, and We'll See You Real Soon!
~Kristen