I think that what you're describing is a "test" to confirm the web sites are running. One way to implement that is to create the web site checker as tests that output their results in JUnit format, then use the JUnit plugin to display the results in Jenkins.
That allows you to run the tests outside Jenkins and see multiple results inside Jenkins.
Each of those techniques have the benefit that you write the test locally, run it locally, then use Jenkins to execute the same test as you run locally.