Maybe – Is anyone testing the third party already? The first thing to do is to find out if the third party service provider is already having a reputable network penetration test provider review the website. If so, due diligence is needed to validate the scope is appropriate, review the methodology, and understand if any key findings were observed. An organization should confirm when it was last tested, when it will next be tested, and if there are any security vulnerabilities that were determined to be tolerable by the hosting provider.
If the third party is not testing the site, or if the testing being performed is not adequate, then yes, the site needs to be tested. Obtain the third party’s permission, as they should be involved in planning, to ensure that the site is tested safely and coordinated appropriately. If the third party won’t allow testing, one should strongly consider obtaining a “right to audit” clause in their contract or locate another hosting provider that accommodates the need for ongoing vulnerability management, including network penetration testing.