A little over a hundred years ago, the 18th amendment of the Constitution was added that prohibited the, “manufacture, sale, or transportation of intoxicating liquors” (Library of Congress). It was not repealed until 1933 with the 21st amendment that then again allowed alcohol to be sold and distributed.
During the era of The Great Gatsby, the Prohibition Era was in full swing, but was clearly not working in major cities through bootleggers, moonshine, and speakeasies. The Great Gatsby without alcohol would be hard to create, and without it, there would be no novel. If anything, there would be no novel if the characters followed the law.

Image from inside a speakeasy during the prohibition.
For starters, Gatsby himself would be nothing without liquor in the novel. It was the way he made his money, and it even if he did have another means of making his money, the cocktails is what drew people to his parties and nightlife. Gatsby would be forced to arguably work harder than he did in the novel, going to a typical 9-5 job and working his way up in postwar society. He would have a more difficult time trying to appeal to Daisy through the way that she likes to be “fun and carefree” but is secretly destructive.
Tom would undoubtedly be better off with following the 18th amendment. He was an aggressive and strong man around people, which is never a good mix with alcohol that we saw clearly in the novel. While out with his mistress, it is safe to assume he was heavily intoxicated when he hits Myrtle right in the face and breaks her nose. Though it is not how he makes his money like Gatsby, he would still be heavily affected by not drinking alcohol. His influence would likely go down in its appeal to lower class women, who probably had less access to quality alcohol, or alcohol at all. Since there was no connection outside of physical between Tom and his other women, it is assumed that all they would do was engage in sex and drink copious amounts without a care. This all comes from what we see in Nick’s interpretation of the apartment in NYC one evening, but I believe it for the most part (even though he was intoxicated too).