Yes, kids can search, add, and edit *online and family rewards from their own device. However, you will still have to approve or deny the request when it comes time to redeem a reward. You can remove or modify their rewards at any time.
*While the online reward function is designed the same way on the kids app as it is on the parent account, we've added filters to their search options. Certain items have been flagged as adult or mature, so those items will not appear in your kids' search. Or, you may simply turn off their ability to search for online rewards in your Family Members setting.