¿Cómo llamar a la API REST de Apex desde Javascript?

Solución:

Finalmente, resolví el problema. Estoy pegando mi código aquí como referencia

    var result = sforce.connection.login("[email protected]", "mypassword+mysecurityToken");
    sforce.connection.init('{!sessionId}', 'REST web service url'); // here pass current session id of the org from which you are making request.

    sforce.connection.remoteFunction({
           url : REST web service url,
           requestHeaders: {"Authorization":"Bearer "+result.sessionId, "Content-Type":"application/json"}, // here pass the session id of the org in which you have your REST service
           requestData: data to post in JSON format,
           method: "POST",
           onSuccess : function(response) {
                  console.log(response);
              },
           onFailure : function(response) {
                  alert("Failed" + response)
              }
       });

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