There were two main Topps printers early on: Lord Baltimore Press (alternatively Lord Baltimore Printing), with offices in New York and a printing plant in Baltimore; and Zabel Brothers Lithographers of Philadelphia, which functioned as Bowman's printer. It's fairly clear to me that, after they acquired Bowman from Connelly Containers in early 1956 Topps, first used Zabel Brothers to print the 1956 U.S. Presidents set. Bowman's main card size at the time of the purchase was of the same height used by Topps but 1/8" lesser in width, which matches the dimensions of U.S. Presidents in '56.
(Lord Baltimore Press logo from 1949 Topps Stop & Go wrapper)
The other Topps Giant Size issues from 1956 (Baseball, Football, Davy Crockett "A" Series, Round Up and Flags of the World) were manufactured in the regular size used by Topps for such cards since 1952 and almost certainly were all the work of Lord Baltimore Press (LBP). Topps went to standard sized cards measuring 2 1/2" x 3 1/2" when Elvis Presley came out in late 1956 and never looked back, so any differences in size were rendered moot.
LBP was purchased by International Paper in 1958 and within about two years had been switched from high quality commercial lithography to producing shipping cartons and the like. I suspect Topps sporadically used Zabel Brothers in the 1957-60 period before switching over somewhat permanently but right now it's impossible to tell. Complicating matters are three other printing firms that come up.
The first of these, Stecher-Traung Lithographers of Rochester, New York may have been involved with the production of the 1962 Baseball green tints series and few other sets. Stecher-Traung also had a facility in Hartford, Connecticut and until about two months ago I never would have associated that state with the printing of Topps cards. However it turns out another printer in the Nutmeg State, namely Chromographic Press, Inc. of Hamden produced some cards from roughly 1966-71 and was owned by Topps director P. Peter Shorin. The plant may have been in New Haven but that's not clear to me right now. The Shorin connection is interesting as Topps co-founder Philip Shorin held at least two patents relating to printing technology.
Then there is the mysterious case of A. Hoen & Company of Baltimore, also commercial lithographers. An obituary for Thomas Townsend Hoen, known as "Townie" and noted to be the last President of A. Hoen & Company appears in the May 25, 2011 issue of the Baltimore Sun and mentions the firm printed cards for Topps. A. Hoen certainly did high quality work as they printed maps for National Geographic. Chromographic Press went into bankruptcy around 1971, likely a planned one as Topps consolidated expenses in advance of their 1972 IPO.
Zabel Brothers was shut down in 1982 after being sold to American Bag & Paper, also known as American Packaging (around 1980) while A. Hoen printed its last in 1981. Stecher-Traung looks like it managed to hang on until about 1985 after it merged with Schmidt Lithographic of San Francisco (printer of the glorious Obak tobacco cards from 1009-11) and an offshoot or two may still be around today. What Topps did post-Zabel I am not 100% clear on. Len Brown mentioned Topps printed their own cards in-house at some point in Duryea while I recall some hobby press articles about high quality printers used by them in the 1990's when the technology really leapfrogged mid-decade.
Large scale commercial lithography of course was done in eventually by more modernized methods of reproduction and printing but despite it being a business, it was also an art form. Just look at the detail in the advertisement above.