How to UPDATE/DELETE/INSERT with Microsoft.ACE.OLEDB.12.0 as Linked Server in SQL Server 2012?
The problem came from an insufficient privileges on the server where the Access file is stored. Here's how you should configure your server depending of the login type that you are using:

If you are using SQL Server login:


Give to the SQL Server Database Engine service account permissions to read and write (depending if you want to make CRUD on that file) on the folder where the Access file reside.


If you are using Windows authentication login:


Give to this user login permissions to read and write :

on the folder where the Access file reside.
on the temp folder of the SQL Server Database Engine user account. For exemple, if the SQL Server Database Engine service account is domain\sqlEngSvc, the read/write permissions should be set to the C:\Users\sqlEngSvc\AppData\Local\Temp folder.