Web API Multiple Authentication during login

We can add multiple authentication middleware directly in the web API project. To add the authentication using Azure AD, we can use Microsoft.Owin.Security.ActiveDirectory. And here is the code support both for the individual and Azure AD account for your reference:

public void ConfigureAuth(IAppBuilder app)
{
    // Configure the db context and user manager to use a single instance per request
    app.CreatePerOwinContext(ApplicationDbContext.Create);
    app.CreatePerOwinContext<ApplicationUserManager>(ApplicationUserManager.Create);

    // Enable the application to use a cookie to store information for the signed in user
    // and to use a cookie to temporarily store information about a user logging in with a third party login provider
    app.UseCookieAuthentication(new CookieAuthenticationOptions());
    app.UseExternalSignInCookie(DefaultAuthenticationTypes.ExternalCookie);

    // Configure the application for OAuth based flow
    PublicClientId = "self";
    OAuthOptions = new OAuthAuthorizationServerOptions
    {
        TokenEndpointPath = new PathString("/Token"),
        Provider = new ApplicationOAuthProvider(PublicClientId),
        AuthorizeEndpointPath = new PathString("/api/Account/ExternalLogin"),
        AccessTokenExpireTimeSpan = TimeSpan.FromDays(14),
        // In production mode set AllowInsecureHttp = false
        AllowInsecureHttp = true
    };

    // Enable the application to use bearer tokens to authenticate users
    app.UseOAuthBearerTokens(OAuthOptions);

    app.UseWindowsAzureActiveDirectoryBearerAuthentication(
      new WindowsAzureActiveDirectoryBearerAuthenticationOptions
      {
          Audience = ConfigurationManager.AppSettings["ida:Audience"],
          Tenant = ConfigurationManager.AppSettings["ida:Tenant"],

      });    
}

To authenticate the local account, we can get the access token from the authorization server build with web API project. And for the Azure AD account, we need to get the token from Azure AD.

How do I configure both. In other words, know which mechanisms to use in the API

In the front-end application, you should also provide a button to login with Azure AD then acquire the access token from Azure AD. Then you can call the web API using this access token as the individual accounts.

Secondly, how do I make use of a different ida:ClientId (Azure AD Auth) during run-time (multi tenant site)

If you want to develop a multi tenant site, when you register the web app/API app on Azure AD, we need to enable the Multi-tenanted. And replace the tenant in the authorization/token endpoint with common. After that, the users from other tenants could login-in your app. More detail about multi-tenant development, you can refer link below:

How to sign in any Azure Active Directory (AD) user using the multi-tenant application pattern