To allow a user to add new Campaign Members or update Member statuses, the administrator should make sure that Marketing User Checkbox is checked on the user record page. This checkbox enables users to create, edit, and delete campaigns, configure advanced campaign setup, import leads, manage campaign members, and update campaign history via mass update. The checkbox also requires users to have Read and Edit permissions on campaigns and leads/contacts. Creating a permission set, providing access to both Leads and Contacts, or running a Campaign report will not enable users to manage Campaign Members. References:
https://help.salesforce.com/s/articleView?id=sf.campaigns_enable.htm&type=5