Is this your corporate iPhone? Do you have any profile installed by your corporate / organisation / enterprise / School Management then please contact the IT system administrator of your corporate / organisation / enterprise / School Management.
If you have installed any profile on behalf of your corporate / organisation / enterprise / School Management then uninstall it and try to add contacts. --> Install or remove configuration profiles on iPhone - Apple ...
Nevertheless...
Contacts are neither stored locally on an iPhone nor in SIM. They are synced to an email account that you had Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other
- How many Contacts account (email client server) you have?
- What is your default Contacts Account?
- Which email account you are using to store Contacts?
- See the GIF below for Contacts and Default Account
- Learn how to add an email account and make it default (Watch the GIF)
- This example is for Gmail but the process remains same for all Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other