Yes, your company need to comply with ISO 9001:2015 clause 8.5.1, independently of who designs the product.
a) need product specifications that describe features and performance targets
b) need the necessary monitoring and measuring resources to confirm product quality according to specifications and performance targets
c) need a process control plan that defines for each manufacturing step what is relevant to control with what frequency and within which limits in the process. (For example, a temperature or pressure control). Need a process control plan that defines for each manufacturing step what kind of quality control should be done for raw-materials and work-in-progress materials with what frequency and within which limits
d) need suitable equipment (to avoid any abnormal rate of breakdowns or lost production time). Need a suitable environment (If relevant to protect product quality – moisture control, pest control, …)
e) need enough people in your production process and they need to be competent according to criteria
g) need to determine risks of human error, evaluate potential consequences and implement actions to prevent it.