I work as a TA full time from home, but corporate rather than leisure (corporate is a whole nuther kettle of fish...) but I do know a bit about the independant leisure side and of course the industry in general.
As a whole the discounts ain't what they used to be, it was going down hill when I started in the late 90's and industry discounts generally just aren't there anymore unless you're a top producer and that usually means FT rather than PT work. As a perk of my position I get a certain number of free airline tickets a year and that's about it. Lets put it this way, my main credit card for reward points is a Starwood Amex so I can get free hotel stays, that's how lousy the discounts have gotten.
That said, it's still a good option for people looking to work from home and/or want a 2nd income. I would not recommend it as a main source of income, it can take a while to start getting paid and like any other sales position, you can't alway be sure what you'll make from month to month. Also no fringe benefits like health care, 401k, etc.
I would also be open to booking destinations other than Disney. Focusing on Disney might be a good way to get your feet wet but I'd really look in to expanding if at all possible. A lot of people feel comfortable booking Disney on their own without the help of an agent, and anyway there are tons of people who (beleive it or not!) have no interest in traveling to Disney, or only want to go once "for the kids". Crazy, I know!
So you'd be missing out on a lot of potential clients right there.
While you don't need to be on call for your clients 24/7, you do need to be available a lot, especially evenings and weekends as that's when your clients are more apt to be calling/emailing you to book or if they have questions. You might be able to do something like you are "closed" monday and tuesday mornings if you need a set time off, but generally you'll need to be available during waking hours. That said, if your host agency has an after hours emergency hotline that would be ideal. You should be getting something for that commission split afterall, if your host doesn't offer one I'd look for one that does.
ETA: My own agency has an after hours emergency number, most large agencies do. It's only for people who are traveling and are having an issue, anyone looking to make a new reservation or has questions about an existing reservation (that they aren't currently traveling on) is told to call back during normal business hours.
When working on your own, getting time off can be tough. You'd either have to leave your clients to their own devices for however long you want to be gone or have someone be able to cover for you.
Also, "working from home" does not automaticly equate to "don't need to pay for child care". I work from home FT and still send my DD to a sitter 2x a week, if it wasn't for my husband working 3rd shift and being willing to short himself sleep, it would be 5x a week just as if I worked at an office. I can't work and watch my DD at the same time, it's not fair to my employer OR my DD.
As other's have pointed out, you'll get a lot of "shoppers". They want a ton of quotes and once you find the best deal, they suddenly aren't interested. Sometimes that's true but often they'll just go and book on their own. There are ways to combat that, like asking for $25 upfront to start searching and then you credit that to them if/when they book but then you can also turn away clients that way too, if they don't want to pony up. A better way to handle it is to keep track of those that want quotes and don't book and do a "3 strike rule", 3rd time they want a quote and don't book, you drop them as a "client' (or charge then the $25 reserch fee as I previously stated). Don't ever be afraid to "fire" a client that doesn't make you any money.
When you aren't researching or booking travel for someone, your job is to read read read and keep up with things. If Disney is your only destination it won't be quite as hard but things at Disney do change and you need to keep up. This Fantasyland expansion is a great example, your clients will want/need to know about it and you have to know what you're talking about, even if you haven't seen it for yourself yet.
Travel agents/agencies are making a bit of a come-back. A lot of people don't want to spend hours trying to reserch trips on Kyak or Expedia and just want someone ele to "deal with it" so you're picking a good time to try it out.