Hello,
Per our documentation, you need to either be an Author, Editor, or Administrator in order to insert new data rows into any table in general.
If you are trying to run code, such as JavaScript or R within LabKey (via wikis or the R View Builder), the user that isn't an Admin will also need the Developer role enabled for their user as well:
https://www.labkey.org/Documentation/wiki-page.view?name=matrixReportPerms
https://www.labkey.org/Documentation/wiki-page.view?name=rolesPermissionsMatrix
If you are using the LabKey API, such as Python, Perl, R (externally), Java, or accessing the HTTP API endpoints, LabKey uses the same permissions for the user, so the person leveraging the APIs would also need to be an Author, Editor, or Administrator.
https://www.labkey.org/Documentation/wiki-page.view?name=viewApis
Regards,
Jon