Absolutely. I have a family of four as well, and we are by no means wealthy. We have taken our kids to Disney multiple times, and we are on our way back again in April (for my 11 yr old this will be her 4th trip, my 7 yr old son his 3rd.) If you budget and plan, a Disney vacation is not impossible.
My first suggestion is to plan the trip yourself. Paying someone else to plan my vacation is someting I would rather not do. I have always made all of our reservations and never had a problem. I have two sources that I always use. First is the Birnbaum's guide to WDW. There are a ton of books out there, but we have always found this one to work the best for us. Second, the Disney website (www.disneyworld.com) has TONS of info on it, right down to an estimate on what your particular package will cost. Sure, you may be able to get deals just about anywhere, but buyer beware! If a deal sounds too good to be true, it usually is. I have had people that I know go with a deal they got and get burned. They stayed on the grounds, but the trip was did not include things that they wanted to do. I have never run into a problem.
Make the reservations as far in advance as possible. Disney takes reservations up to 18 months in advance, I believe. You will not get the final price of the vacation that far in advance, because the prices for the following year usually don't come out until the fall. However, you can make payments to your vacation, either monthly, quarterly, whatever. When you are ready to go, your entire trip will be paid for, except for spending money.
STAY ON THE GROUNDS. Disney has value and moderate resorts that are very competitve in terms of price and the service is outstanding. You can get a package through Disney that includes room and tickets for everyone to the parks for a great price. I always book my airfare separately, just because I am weird, but Disney can do that for you too. By staying on the grounds, you get to take advantage of the perks reserved for resort guests only. They have extra magic hours where one of the parks is open early and one stays open late. We always take advantage of the park staying open late. The most popular rides stay open, and the lines are very short. Also, the park is never closed to resort guests, even if they temporarily close admission due to the size of the crowd. You also get unlimited use of the Disney transportation system. They pick you up and take you to the park, then bring you back. My kids love riding the monorail.
Get the park hopper option on your tickets. Many people don't want to spend the extra money for the park hopper option, but it is essential to have a good experience. It gives you the flexiblility to pretty much come and go as you please. You can go to one park in the morning, go back in the afternoon for a rest, then hit a different park in the evening. I truly think that is the best way to go when going to Disney.
For the future you may want to get some info on the Disney Vacation Club. My wife and I joined the vacation club in 2005, and it is the best thing we ever did. Being in the club gives us even more options and flexibility and it is affordable. The whole thing is not like a true "timeshare", but they give you points to spend. You can bank points, borrow points from the upcoming year, and use what you need to go. The points are also usable at around 500 or so other non-disney hotels, so you are not stuck using points to only go to Disney. For example, my wife and I purchased 160 points. I would never be able to get a room for the period of time that we are going to be there for what I am paying for my points. What it does is I am basically saving up for vacation all of the time, and even when the payments are done, I still have the points to use. For the vacation that I am taking in April, we are staying at the Saratoga Springs resort. I used the 160 points from last year, 160 from this year, and a few from next year for a 9 day stay. My room is completely paid for. I get discounted park passes, and discounts all over WDW with my membership card. I only have to pay for my transportation, park passes (discounted), and spending cash, which I saved up for. What this also does is give me the opportunity to stay at some of the resorts that I could never afford otherwise. My kids are begging to stay at the Animal Kingdom lodge. They want to be in the Savanah rooms where you can look out and see the animals. Without the points, I probably could never swing that.
No matter what you do or spend, a Disney trip is worth every penny. The look on my kids faces the first time they stood there and saw Cinderella's castle was worth the cost. The looks on their faces when I told them I made reservations this time was even better. Do your research and plan, and you can give the kids an experience they will never forget.
Disney vacation club member and frequent visitor