In the frontend approval is not possible, you need the backend for that. People can be given the specific minimal rights to be able to just approve bookings for their events (e.g. if they are a contact person for that event + the EME right to approve bookings for their events).
See the EME settings, tab “Access”. You need to assign “Author approve bookings” the capability of choice and then see if a user has that capability. Like indicated in that page, use “User Role Editor” to manage your own roles and capabilities.