Almost all of the journals for psychology for the last 30 years and older are online. Use Library Search Journals when you know the name of the journal to find out which database provides access to it, or if the library does not have access.
Some much older journals and some current journals are not immediately accessible online. If you are looking for a particular article in one of these, put in a request to have the article scanned and sent to you through ILLiad.
If you know the journal comes from the American Psychological Association, you can use the database APA PsycArticles -- although indexing is better in the full database APA PsycInfo.