Can you? Probably. Should you? You may want to give it some thought. With the restrictions the TSA has in effect, you won't be able to carry any liquids, gels, or creams that are larger than 3.4 oz. That can be an issue for a lot of people. I have an Owners Locker at WDW...that has all my liquids, gels, etc. I don't need to bring shampoo, conditioner, deoderant, toothpaste, sunscreen, detergent, etc with me anymore. It's all in that Locker, waiting for me upon checkin.
If traveling in Dec, you are going to need heavier clothing as well. It can get chilly at WDW in Dec...especially at night. I usually fly in one pair of jeans, pack a second pair as well as a few pair of capris. Then, you'll need a sweatshirt or jacket, longer sleeved shirts. Those clothes take up more space than shorts and tank tops.
I would seriously consider one of several options...
1. fly SW if at all possible. You get two checked bags, free.
2. fly Jetblue..you get one checked bag free, per person
3. check one bag, if flying on an airline that charges for that first bag. Pack all that non-allowable stuff in it, as well as heavier clothing. Then pack the rest of your stuff in carryon bags. Each paying passenger gets two pieces...one piece of smaller luggage (must fit the dimensions put out by the airline) and a personal piece...tote bag, purse, etc.
I don't think I would try to fly with just carryon, with a toddler, in Dec.