Dear ChasityS1 ,
Thanks for choosing Microsoft community!
I have read your question carefully and done some research, please kindly refer to following:
At first, there is no cancel booking option when your client click on "Manage Booking" in the confirmation email.
We are sorry for the inconvenience, as a workaround, please let her tries to cancel an appoinment in the Bookings web APP by double clicking the appointment in the calendar and then click on the option cancel booking see below picture:

I think we should have been editing the appointment and clicking 'Cancel Booking'. This then sends an email to our staff member and the customer. Another thing we didn't notice was the 'Remove from Calendar' button which is a manual thing we need to do to remove the cancelled booking from our individual calendars.
Is there a confirmation for the cancelled email? And please make it sure the booking event had been cancelled.
Please let the user login the account to Outlook Web App check whether the event is existing. If yes, is there any error message?
If the event appears in the Outlook client, not website. Please go to Send /Receive section> click Update Folder to sync manually.
Hope this help you, if there is any deviation in my understanding, please forgive me and provide me with more details. I am more than happy to continue to help you. I wish you all the best!
Best regards,
Kerry