When using the Data Import Wizard to create records, the error "You do not have sufficient permissions to import custom objects" may occur due to permissions but also in the following scenarios:
Steps to reproduce:
1) Setup | Data Management | Data Import Wizard
2) Click Launch Wizard!
3) Under the 'What kind of data are you importing?', select Custom Objects
4) You see the error message described.
To resolve the error message, you need the 'Import Custom Objects' User Permission and at least Edit access on the custom object(s) that you are trying to import. See What permissions do I need to import records for more details.
Note: Review Manage Object Access in Object Manager to see the objects to which you have access.
Import Data with the Data Import Wizard
Privileges error when you import custom objects in Salesforce Essentials
000381585

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