In most cases, yes. You can take your purse, backpack, tote or other bag with you on all Walt Disney World rides. In most cases, you can store this item at your feet, even on the thrill rides such as Space Mountain, Expedition Everest, Big Thunder Mountain, etc.
On Kali River Rapids, free lockers are provided and all bags should be placed in these lockers to keep items from getting wet.
We do recommend you try to limit the size of your bag or backpack to be more comfortable on these rides. Since you do have to store them at your feet, some rides, such as Space Mountain, do not leave you much space.