Wikipedia: List of online newspaper archives
Many lack "official" interfaces and may restrict the use of materials (see their copyrights), but those that fit the requirements can be scraped with web services like import.io or Python's Scrapy:
http://en.wikipedia.org/wiki/Wikipedia:List_of_online_newspaper_archives
(It also includes Google's scans of old newspapers.)
Library of Congress: Newspaper Archives, Indexes & Morgues
The links to a few dozens of newspaper archives with full-text articles. International newspaper archives included. Mostly historical data.
All Digitized Newspapers 1836-1922 by state, ethnicity, language:
Their API: