Qualification is the decision that a supplier is fit to supply. Clause 8.4 does not say how you decide - an audit, a sample run, a certificate review - only that you decided on criteria you set in advance and can show what you based it on.
Recording a qualification needs manage_supplier_evaluations. Reading the register needs only read_suppliers.
Recording one
A qualification carries an outcome and the evidence that supports it. A passed qualification with no document behind it is an opinion with a date on it.
Open the supplier and record a qualification.
Say what kind it is: an initial qualification, or a re-evaluation of someone already approved.
Attach what you based the decision on - the audit report, the sample results, the certificate.
Record the outcome, and let it move the supplier's state if that is what the outcome means.
The trail
Every qualification stays, including the failed ones. A supplier who failed once and passed on re-audit is a different supplier from one who passed first time, and the register should be able to tell you which you are dealing with.
Deleting a failed qualification to tidy the record is the one thing that turns this module from evidence into decoration.
Certifications
Certificates a supplier holds - ISO 9001, ISO 14001, a customer approval - sit on the supplier itself rather than on one qualification round, because they outlive it.
Next: score supplier performance.