Hi,
Based on the error, it appears that the security Role assigned to the user doesn't have the Privilege to Share Activities. This is a required privilege to create any activity. Please follow the steps below to resolve the above error.
1) Log-in to CRM as Administrator.
2) Navigate to Settings -> Business Management -> Security Roles.
3) Open the Security Role assigned to the User who is getting the above error.
4) Navigate to the Core Records Tab and provide at least User Level access to Share for Activity.
5) Save the changes.
Clear the Internet Explorer cache after performing the above steps and you should be able to create an activity.
For more details about Security Role UI to Privilege Matching, please refer to the MSDDN link below.