If you're not receiving approval emails when a client submits a form, you may need to change (verify) the Email ID of the Site Guest User profile. Please follow the below steps:
1. Go to Setup and enter "sites" in the Quick Find box, then select "Sites".
2. From the list of sites, click on the label of the site that you created while configuring the Update Client Details app.
3. On the Site Details page, click on "Public Access Settings".
4. It opens the profile page, on the Profile page click on "View Users".
5. On the Users page, click on "Edit" on the left of the Site Guest User profile.
6. Edit the Email field on the Site Guest User profile and put in a valid Email ID from which you want to receive the mails f and hit Save.
Now you will receive emails like the one below when a client submits forms for approval.
PS: Also, make sure the org’s email deliverability is set to all emails. https://help.salesforce.com/s/articleView?id=sf.data_sandbox_email_deliverability.htm&type=5