Yes, it is a common practice to sell daughters [and sometimes, wives]. It is widely practiced at least until the mid of Showa period (1950s) so I think it is safe to make it as a major plot line in 1930s-1940s Japan.
In the Edo period (1603-1868) the practice is extremely rampant both in urban and rural areas, and it is considered normal as it is the zeitgeist of the time. However, during the Bakumatsu (opening of Japan) in the early 19th century, Western influence is getting stronger, and there are protests to this practice.
Most protests are elite-driven (from the Westerners and the Western-educated Japanese) though, and even the government legislated Ordinance no. 44 in 1900, daughters [and wives] are still being sold. Especially families in rural areas. There were people that was called Karayuki-san (which means, Ms-Gone-to-China). They were being sold to Asia prostitution traffic - mostly went to China, hence the name.