OCSP stapling is an extension to the TLS protocol which is defined in RFC 6066 and extended in RFC 6961. The client announces support OCSP stapling to the server during the TLS handshake and if the server has a OCSP response for stapling it will be send as CertificateStatus with the TLS handshake.
It does not matter who fills the role of client and server: The server might be a web server supporting TLS and the client might be a web browser as in your question. Or the server might be a mail server supporting TLS and the client a mail client. Or the server might be Bob and the client Alice, speaking their own protocol protected by TLS. All it matters is that both sides use TLS for communication and that the client requests OCSP stapling and the server provides the OCSP response within CertificateStatus.