There are few welding codes that require the involvement of a CWI. However, most welding standards and codes do require the welder to be qualified by testing. It is typical for the welding standard to make the employer responsible to ensure the welder is properly qualified by signing (i.e., certifying) the performance/qualification test record.
The employer can typically qualify the welder in-house if they have the ability to do so or they can use the services of a third party, i.e., a CWI, to help them with the task. The pitfall is still the employer's responsibility to ensure the welder is qualified per the requirements of the applicable welding standard. To say it another way, many welding standards do not recognize the qualification/certification performed by the third party. It is the employer that must sign (certify) the test record is correct.
Contrary to some advertisement, welder qualification and certifications performed by third party laboratories is not universally transferable from one employer to another unless it is explicitly permitted by the welding code governing the work. Even the AWS structural welding code/steel gives the Owner the prerogative to accept previous qualifications by another employer or third party. Generally, work performed to ASME does not recognize qualification/certification by a previous employer or third party. That being said, there are circumstances and conditions where it is permitted. ASME B31.9 comes to mind. In that case, I believe the current employer is required to obtain a copy of the original test record and the current employer must sign the certifying statement.
The CWI is required to perform the visual inspections of the welds when it is stipulated by the contract. So, while it may not be a code requirement, it may well be a contractual obligation. While the code may not require the visual examination be performed by a CWI, most require the inspector's qualification be documented, reviewed and approved by the Owner (usually through the Engineer representing the Owner).
If you provide more information regarding the applicable code invoked by your contract or specific language included in the project specification, a more explicit response may be offered.
Best regards - Al