As the film is set in the 1920s, it begs the question on whether it was it legal for Mormons to practice polygamy in the 1920s. Was Ezekiel Young breaking US law by having six wives?

No. In fact there was virtually no point at which Mormon Polygamy was practiced legally. It was illegal in Ohio when Joseph Smith took his first plural wife. It was Illegal in Illinois when the leadership started taking up the practice in earnest in the early 1840s. It was legal in US territories when the Mormons first moved to Utah. But that was fairly quickly made illegal. So they had about a decade where it existed in a legal limbo, neither approved, nor banned. But that was taken care of by 1862.

Did the LDS church abolish the doctrine of polygamy by this point? If not, when did they officially stop practicing polygamy?

Well now there is an interesting story. For a long time the US basically winked at Mormon polygamy. They were in a far off and isolated territory, and they had bigger things to worry about... like the US Civil war. None the less polygamy was one of the "twin relics of barbarism" that had been condemned alongside slavery in the Republican platform since 1856. In 1862 the Morrill Anti-Bigamy Act was passed. It banned polygamy in all US territories and also limited how much property a church could own... clearly aimed at the LDS Church. Lincoln however sent notes to General Patrick Edward Connor at Fort Douglas explicitly telling him not to enforce the law in the Utah Territory, so long as Brigham Young otherwise behaved himself. Lincoln didn't want to have send additional troops to Utah. To him the Civil War took precedence.

The Morrill Act thus went largely unenforced. In 1882 congress passed the Edmunds Anti-Polygamy Act, which made simple cohabitation illegal, nominally making keeping a mistress illegal. this was intended to get around the fact that these polygamist marriages were never registered and thus were never "marriages" per se. It also revoked polygamists' right to vote, to serve on juries, or hold political office. In 1887 congress finally got serious. It passed the Edmunds–Tucker Act. That dis-incorporated the LDS Church and the Perpetual Emigration Fund, confiscated most of their property largely disenfranchised Mormons, and reappointed most territorial officers and judges. It was a hammer blow to the LDS Church. Enforcement of all the above suddenly started happening regularly. Within short order most of the church leadership was either in jail or in hiding. In 1889 the Supreme Court ruled on the acts and upholds them.

So in 1890 President of the LDS Church, Wilford Woodruff, releases what is called The Manifesto, declaring that the LDS Church declaring that the church has ceased to practice polygamy. In fact it continued the practice. Apostle John W Taylor married his 3rd wife a week after the Manifesto was released. Around the same time a group of Mormon colonies were established in Canada and Mexico to preserve polygamy should things in the US get worse. Polygamy is continued for years. In 1897 Woodruff married his 11th wife, at the age of 90; she was 40. In 1898 Lorenzo Snow became the church president and was determined to stop new marriages within the church. In 1900 he released an additional Manifesto absolutely banning polygamy. However new marriages continue. At least three apostles took additional wives in 1900 and 1901. All three claimed they had Snow's explicit approval.

In 1901 Joseph F Smith (Nephew to the church founder) became the prophet. He releases yet another Manifesto against polygamy. But the marriages continue to 1905. The Apostle Matthias Cowley has the last officially approved marriage. He later resigns as an apostle and is briefly disfellowshiped. When his wife Lenora Taylor dies in 1971, she died as the last officially recognized polygamist. However by now the leadership had realized the US wasn't going to back down and allow them to practice in secret. Smith eventually started excommunicating anyone performing new marriages. However they never asked anyone to leave existing polygamist marriages. So there are plenty of Mormons who lived openly with their wives up till the 1940s and 1950s. In general, once new marriages largely ceased, the government stopped prosecuting existing polygamists, letting the practice to die out naturally.

There are also secret groups within the church who continue to marry in secret from both the government and the main LDS Church. The practices was insignificant the government had largely quit seeking prosecutions. But the main church starts excommunicating members who are found out. The biggest name is the Apostle Richard Lyman who was excommunicated in 1943 for a secret marriage performed in 1925. These secret groups mostly eventually peeled off between the 1950s and 1980s to form breakaway churches such as the FLDS.

So to answer your question, the LDS church said they stopped new marriages in 1890, officially internally stopped them in 1905, and in actuality still haven't completely stopped polygamist marriages in the church. There were plenty of people living polygamy in the western US in 1920. An Ezekiel Young as mentioned in Chicago would have been a disappearing breed, but still quite plausible.