I sent my application on 17/03/2020 by post, on the 90th day of my stay based on visa-free travel. Due to the closing of the service, I cannot get a stamp in my passport. Can I work without a "stamp"?

Yes, if the last deadline for submitting your application was during an epidemic emergency/epidemic state and you have already applied for a temporary residence and work permit, your stay is considered legal. If you hold a valid work permit or a declaration of entrusting a work, you can continue your employment with the same employer until the end of the procedure.