No, clients cannot add Categories to their Project. Only you or your team can add, edit, or remove categories from a Project.
As long as you've given them permission, a client can login to their portal and do things like: add a task, add a vendor, create a mood board, etc. They have the option to link those items to an existing category (if applicable). For example...
Your client booked a string quartet and wants to add their details to the project. They login to their portal and click Add Vendor. When adding the vendor details, there will be a category dropdown populated with the categories you've already added to the project. They can select one of them. Next to the dropdown is a notice: Need a category that's not on this list? Reach out to your planner to get it added.