Certainly GPL2 and GPL3 (and the LGPL versions) allow you to use the code in your web app or web service and not distribute the source. The MIT, BSD, Apache, and other similar permissive licenses also allow it.
The AGPL does require that you distribute the source code of your derivative web app or web service. Section 13 requires that you provide the modified source to users that interact with the system remotely.
That said if you are making money on the app it is worth talking to a lawyer to be sure. If no money is involved I would be less concerned about getting a lawyer involved.
Also, you can often avoid a lawyer by just asking the authors of the libraries you want to use.
