Access FBA SharePoint site using web service

You can access any site collection programmatically   as below code 

You have to use Authentication.asmx web service for authenticate any user.

Then you can access any lists of this site.

// /_vti_bin/Authenticate.asmx web service's Object
            Authentication.Authentication objAuthentication = new Authentication.Authentication();
            objAuthentication.CookieContainer = new System.Net.CookieContainer();
            LoginResult loginResule = objAuthentication.Login("username", "password");

            if (loginResule.ErrorCode == LoginErrorCode.NoError)
            {
                CookieCollection objCookieCollection = objAuthentication.CookieContainer.GetCookies(new Uri(objAuthentication.Url));
                Cookie authCookie = objCookieCollection[loginResule.CookieName];
                
                // /_vti_bin/Lists.asmx web service's Object
                Lists.Lists objLists = new Lists.Lists();
                objLists.CookieContainer = new CookieContainer();
                objLists.CookieContainer.Add(authCookie);

                System.Xml.XmlNode objListXMLNode = objLists.GetList("Shared Documents");
            }

Happy Coding !!!