My professor says I can't use Web sources for my paper.
Ask your teacher to clarify this, because there are basically two Webs. There is the public Web that search engines find and everybody can use free of charge. Then there's the invisible or "deep" Web, content that is located on the Web but can only be viewed by subscribers, like the periodical databases the Library provides for you.
An article you find using an online periodical database such as ProQuest Research Library was first published in a journal or magazine in print. It's just delivered to you via the World Wide Web. This is an example of the invisible or "deep" Web.
Take a moment to read about the difference between the Web and online databases.