Hi,
“As an obsessive person, you realize that there is a better version of everything out there,” said Sean Brock, the Charleston chef whose restaurant Husk serves only food produced south of the Mason-Dixon line, from Georgia olive oil to Tennessee chocolate to capers made from locally foraged elderberries. Many Southern chefs are working along similar lines — Frank Stitt, Mike Lata, Andrea Reusing and Linton Hopkins are just a few — but Mr. Brock’s rigor has redefined what it means to cook like a Southerner today. Young chefs are joining in, learning butchery and fermentation, putting up chowchow and piccalilli, experimenting with wood ash to make their own hominy.
Perhaps most important, they are paying (and charging) big-city prices for down-home ingredients: money that is keeping food traditions, and small producers, alive.
(Southern Farmers Vanquish the Clichés. The NY Times)
Does "putting up" mean "can, preserve" in the context above?
"Chow chow" is apparently a pickled relish, so yes, I would say that is what is meant.
Yes, "putting up" is a common AmE phrase used to mean "can" or "preserve" fruits or vegetables for future use.
"I put up eight quarts of pickles this year. Last year we had bad weather all summer and only managed to grow enough good cucumbers for four quarts."
The metrically-minded may need this link: Quart - Wikipedia, the free encyclopedia
Context is always important; labelling is rarely important.