From the main library page (foley.gonzaga.edu), click on Periodicals@Foley, and search for the title of the journal you are looking for. Then click where it says "Click links below for access options."
As you can see, the ASOR Bulletin is available in a bunch of our databases, with different age ranges. This can look confusing until you know what you're looking at:
We can see that JSTOR has the biggest collection of full text for this journal, going back all the way to its founding in 1921, but that it always lacks the last four years (this is called a rolling embargo, and you'll find it in a lot of databases — publishers set it so they can sell current access to a journal apart from its archives). But Academic Search Complete and the Religion and Philosophy Collection both have the full text from 2001 to the present. Combined with JSTOR, that means Foley has the entire run of this journal, from 1921 to the present day!
If you click on the links, you'll be taken to the journal's page in that database:
Different databases have different looks for this, but they will all include those two features: search within the publication, and browse by date.