How can I obtain my client's authorization to manage their claim on their behalf?

There is no specific procedure for this. This is a matter between you and your client.

However, we would like to draw your attention to GDPR requirements.
By accessing your client's customer portal, you may also gain access to personal data that forms part of their private life (such as medical records, for example).
Therefore, it is advisable to obtain your client's explicit consent before managing their claim on their behalf.