The registration follows the legal entity and the site, so a stock sale generally preserves it while an asset sale requires the registrar and the customers to recognize the new entity.
More detail
The mechanics are administrative but the timing is not trivial. In an asset purchase the buyer typically needs a transfer audit or a new certification against the existing quality system, and customers who approved the supplier will re-approve under the new entity. In a stock purchase the certificate stays with the entity, though notifications are still required and customers still re-approve. In aerospace and defense the customer approvals matter more than the registrar's certificate, and prime customers set their own re-approval timelines, which is why the sequence of notification, audit, and re-approval gets written into the purchase agreement rather than left to chance. A production interruption caused by a lapsed approval is the outcome both sides are trying to avoid.