Vinson himself was undecided after it was first heard, but died before the rehearing was held. Vinson himself said: "I am not sure what we should do here today". I know Stanley Reed almost voted to uphold school segregation, but Earl Warren convinced Reed at the last minute. What would the outcome have been if Vinson hadn't died?
The more interesting counterfactual is if Truman appointed James Byrnes instead of Fred Vinson.
Vinson was a well-qualified appointee, in that he had experience in the executive (serving Secretary of the Treasury), legislative (former Congressman from Kentucky), and judicial (former circuit court justice) branches. However James Byrnes was another Truman cabinet member who was strictly better qualified. He was the serving Secretary of State, had served as a Senator from South Carolina, and had been a Supreme Court Justice. Why Truman picked Vinson is somewhat unclear, and may have been the result of personal tensions between Byrnes and Truman, as Byrnes had viewed himself as Roosevelt's heir apparent.
In any case, Truman easily could have chosen Byrnes, and he would have been approved by voice vote. In our world, Byrnes lived until 1972.
As a white South Carolinian born in the 19th century, he was a committed segregationist, and was in fact one of the parties being sued in Brown v. Board (he became the governor of South Carolina in 1950). There's a chance he could have convinced Frankfurter to oppose the decision. In any case, a 5-4 decision with a dissent from the sitting Chief Justice would be far harder to enforce than an unanimous decision, especially given Eisenhower's own doubts.
Eisenhower's promise to Warren would lead to Warren replacing Jackson when he passed, so the likelihood of something like Brown v. Board being issued would be high. Nonetheless, Warren would be the most junior Justice, rather than the Chief.
The liberal turn in American public opinion on racial issues during the 50s and 60s was not entirely the result of the court, and something like that probably would have happened regardless of the influence of one man. Still, the Chief Justice is a powerful office, and Byrnes would have fought a number of rearguard actions.
Baker v. Carr & Reynolds v. Sims (which dealt with state legislative districts which were not equipopulous), probably would have been decided the other way. Miranda v. Arizona almost certainly would have. A 5-4 decision in Brown v. Board would delay any further efforts along those lines.
The impact that this would have on electoral politics is unclear.
