No. Under Article 10 of Law No. 6 of 2010, it is explicitly prohibited for any employer to employ a foreign national without first obtaining authorisation (a work permit) from the competent authority — which is the Public Authority for Manpower (established under Article 9). This applies to all expatriate employees in Kuwait's private sector.
The employer is responsible for completing the required procedures, submitting the necessary documents, and paying the applicable fees as specified by the Minister. If an application for a work permit is refused, the decision must be a reasoned one. Non-compliance by the employer can result in legal penalties.
As an expat, you should always ensure your employer has secured your work permit before you begin working. Working without proper authorisation puts both you and your employer at legal risk, and could affect your residency status in Kuwait. If your employer asks you to start work before the permit is processed, request written confirmation of the application's status. Never accept verbal assurances alone.
This is general legal information, not legal advice. For advice on your specific situation, consult a lawyer licensed in Kuwait.