Hotel options for Disney World abound. You can stay at a Disney resort, you can stay on property at the Swan or Dolphin, which get Disney hotel perks but aren't operated by Disney, you can stay on property at other hotels, or you can stay off property, including at a Good Neighbor Hotel. Each of these options will have ways to book using points, but we're going to focus on the first two because they're the most appealing for Disney travelers.
Booking the Swan and Dolphin with Points
Booking a stay at the Swan or Dolphin with points is your most straightforward option of getting a Disney experience with points. That's because those hotels are actually owned and operated by Starwood, now a part of Marriott. This is, for example, why you'll book room-only bookings through the Swan and Dolphin websites, and not the Disney World website.
Experiences may vary on the Swan and Dolphin websites, especially if the merger with Marriott changes anything, but typically the site will show you how many points per night a booking costs, with average nightly rates tending to range between 8,000 and 12,000 Starpoints.
To get those Starpoints, your best option is going to be the SPG Amex, which currently offers 25,000 Starpoints as a signup bonus for $3,000 in purchases the first 3 months. Alternatively or additionally, the Marriott Rewards Premier credit card from Chase gets 80,000 Marriott points (transferrable at a 3:1 ratio to 26,667 Starpoints). If you and a partner can each get both those cards, you'll have over 100,000 Starpoints to work with. That's up to 15 nights!
Booking Other Disney Properties with Points
The other way to book Disney hotels with points is to use the Chase Ultimate Rewards travel portal. You can earn Chase UR points from a variety of sources, the easiest being the signup bonuses on the Chase Sapphire Preferred and Reserve cards, each currently at 50,000 points for $4,000 spend in 3 months.
Once you have those UR points, you have a bit more flexibility than with Starpoints. We have a post on booking Disney hotels with UR points, but the short of it is that the Chase Ultimate Rewards program has its own travel portal where you can book a wide variety of hotels, including some Disney properties. If you have the Chase Sapphire Preferred, you'll get 1.25 cents per UR point on these bookings. If you have the Chase Sapphire Reserve, you'll get 1.5 cents per UR point on these bookings.
The Chase portal varies in which hotels it shows, and it offers the non-discounted rate at Disney hotels. If there is an ongoing discount, or if you can find a good discounted rate elsewhere, using the UR portal might not be your best option (but bear in mind that 1.5 cents per UR point is a pretty good value).
There's nothing difficult about booking a Disney hotel using points. Your options won't always be perfect, but if you can get the required Starpoints or UR points (which is easy through signup bonuses), you'll have plenty of options to book your Disney stay entirely on points.