I would like to offer a customer a cloud service in form of a website. Now I had the idea that it would increase security enormously if this customer (several employees) also authenticated himself with a client certificate. This certificate should then be installed on his office computers and tablets. The server SSL certificate is a Letsencrypt certificate. From the client certificate I would then also know the private key, but from my point of view this is not dangerous because it is only used to authenticate against my server.
Is there a practicable solution to provide the customer with a client certificate whose public key I then store on my HTTP server?
Can I buy such client certificates from an authentication authority? Did I understand correctly that if I generate a own client certificate, the customer also has to trust my CA and this represents a security risk? Or can the customer only trust this one generated certificate and not my self signed CA/Root CA?
Thanks for anwsers