The Zeitschrift für mathematische Logik und Grundlagen der Mathematik was one of the few logic journals around in the mid 20th century. It started publishing in 1955, I think the only logic journals that are older than it are the Journal of Symbolic Logic (1936), the Archiv für mathematische Logik und Grundlagenforschung (1950) and Studia Logica (1953). It didn’t help that it used to be published in the GDR, and so getting copies of old papers from it has always been a minor hassle if you weren’t lucky enough to be near a library with a subscription going back far enough. All the more exciting that I noticed yesterday that Wiley, who owns the successor journal Mathematical Logic Quarterly, is making the complete backset available online. The others are also available back to vol. 1–the JSL through JSTOR, Studia Logica and the Archiv through Springer. Volumes 1-26 of the Archiv are also available free at the GDZ.
I’m not sure if I should count it as a logic journal, but I’d be remiss if I didn’t also note that Fundamenta Mathematicae has been around since 1920, and back issues are available (free) through the Polish Virtual Mathematics Library.