If you are implementing an LWR Salesforce Experiences Cloud and there is a request to replace the Standard login page using a custom login page, then perhaps this little guide would help you. Assuming you have knowledge of how to set up Salesforce SSO.

You can create an APEX Controller, use your LWC to call the method to receive the link, and then redirect to SAML with a signed SAML Request. By using Auth.AuthConfiguration.getSamlSsoUrl method

string communityUrl = 'your community url';
string startUrl = 'your start url usually /';
string samlId = 'Id of you SSO setup';
String requestSAMLURL = Auth.AuthConfiguration.getSamlSsoUrl(communityUrl, startUrl, samlId);