Hi there Whitney, and welcome to the Disney Parks Moms Panel!
Like you, our days with a stroller are long gone. Such a great place to stow stuff you don't feel like carrying, and for our son to take a little snooze while Mom and Dad enjoyed a quiet moment. But it is awfully nice that the little guy is jumping on the thrill rides with us now, too!
Just about any kind of bag or purse is allowed in the Theme Parks. According to the Park Rules and Regulations, the size limit on bags is 24" long x 15" wide x 18" high. That's about the size of a suitcase, so your normal handbag, tote bag or backpack will be just fine.
There also aren't any attractions at Walt Disney World that require you to ride without a bag. Many will have a compartment onboard where you must stow your bag while you ride, but you don't have to leave it behind before boarding. I have two different backpacks I carry, one small and one large, that I switch out depending on what I need to bring along that day, and I've never had any problems carrying either of them on any attraction in the Theme Parks.
I hope this helps with your planning, Whitney. Please come see us again if you have any more questions. Helping Guests have practically perfect Walt Disney World vacations is what we're here for!
Keep Disney Dreaming,
Kathy K.