Use Journal Finder to see which journals you can get. Some titles are within the High Library itself (in print, microfiche, or microfilm), others are available electronically in various databases. If the journal is held, check to see if the dates you need are available. This is a powerful gateway to about 33,000 titles!
If the periodical you want is in microfilm or microfiche, you can order copies of the articles online by using ILLiad. Click on Request an Article and fill out the form. You will receive an email with a link when the article is available, usually within 24-48 hours (M-F).
Articles that are not available either in the library or in library databases can be ordered on interlibrary loan, through ILLiad. ILL can take anywhere from 3-5 business days, on average.