They went to the store and bought it!

Apparently soy sauce was one of the things incoming immigrants brought with them, but of course that supply wouldn't last. Importing foodstuffs and cookware from China was affordable thanks to bulk shipping in two ways. First, there were a lot of people buying the products. Second, they didn't just buy soy sauce. They bought the material goods of South Chinese culture: shark fins, cassia, Chinese potatoes, rice, coffee, preserved fruit, salted seeds, Chinese pickles, assorted fish, peanut oil silk, firecrackers, porcelain, lacquered dinnerware, spices, Chinese paper and India ink for writing, what have you.

Import-export firms on both sides of the Pacific organized quickly as eager store proprieters in, above all, San Francisco reached out. By 1856, there were 33 stores selling "general goods and groceries" in San Francisco's Chinese neighborhoods. From there, Chinese and Chinese-American people working in parts distant could--and did--order their products. The same year, missionary William Speer observed miners in the California hills buying Chinese rice, tea, star anise, Chinese candy, and of course soy sauce (though he seems to imply they sourced most of their meat, vegetables, and non-rice grains locally). In the 1860s, miners in Idaho were storing imported pickles in special ceramic pots and cooking with jars of soy sauce.

By maintaining so much of their native foodways, Chinese immigrants and their descendants made it affordable to import while saving money sourcing what they could locally. And then, on wagons and railroad cars that followed railroad and mining camps across the West, they set up restaurants serving Western-style food to white workers at a markup.