Yes, but few soldiers actually voted, either due to restrictions imposed by their home states or confusing voting rules designed to make it difficult to tell whether an individual soldier was eligible to vote. This was by design and made it logistically impossible for the military to deliver ballots to soldiers. Many states (particularly in the South but also out west in states like Arizona) had very restrictive voting laws designed to prevent African Americans and other minorities from voting. The result was that for soldiers to be able to vote, the military had to produce 48 different ballots and expend time and energy figuring out which ballot to give each soldier and to determine whether that soldier had to right to vote in the first place. Not a recipe for wide voter participation among 11 million soldiers serving abroad in locations around the world. In fact, in 1944, only about 85,000 soldiers serving abroad voted.

FDR did try to push for a simple federal ballot for soldiers to use. His thinking was that if it was easy for soldiers to vote, they would vote in large numbers for their commander in chief in a time of war. But not just that, if Americans were fighting for democracy, it made sense those soldiers putting their lives on the line should get to vote as well.

But his proposal was feared by both Southern Democrats (who didn't want blacks voting under any circumstance even if they were fighting for democracy in Europe and Asia) and Northern Republicans (who feared liberal voting rules would ensure the re-election of FDR) and both groups opposed the changes. Southerners stated that liberalized voting laws would threaten white supremacy, led to miscegenation and threaten the safety of white women in the South. If black soldiers were able to vote abroad, what would stop them from voting once they got home?

Northern Republicans based their opposition on the idea of states' rights, basically the federal government had no business setting voting standards for the individual states. Republicans openly feared that Roosevelt would simply order soldiers to vote for him.

The compromise bill, passed in 1944, created a uniform federal ballot. But in order for a service member to obtain such a ballot, he must comply with detailed instructions and confirm he was eligible for vote and he had to ask for a state ballot first. If the state ballot had not arrived by Oct. 1, 1944, the soldier received a Federal ballot. However, the state in which the service member qualified to vote in must first have passed legislation to authorize use of the shortened Federal ballot. About 17 states passed bills stating their ballots were adequate and therefore no federal ballot could be used, another 18 stated that soldiers could use a federal ballot, while some other states had laws that were vague. Needless to say, very few soldiers voted in the 1944 presidential election.

You can see two trends become emerge in this fight over the right for soldiers to vote. First, you can see the beginnings of the modern civil rights movement, which had been pressuring FDR to liberalize voting rules for soldiers, pass an anti-lynching act and eliminate discrimination in government hiring (including desegregating the federal government and the military) since the start the New Deal (this was only part of the emerging civil rights movement, which attacked Jim Crow and discrimination from many angles. The NAACP Legal Defense fund, for example, was litigating a series of challenges to segregation in education, starting with law schools and graduate schools.). Black political power grew, in part, because of a mass migration from Southern states where they could not vote to urban cities in the North where they could vote. These black voters generally joined the existing Democratic machine in Northern cities (even though in the South, many blacks would continue to be Republicans). This migration would become a veritable exodus in the postwar era. FDR was only able to manage an executive order banning discrimination in military contracting. Truman would end discrimination in the military and add a civil rights plank to the Democratic party platform. Congress would never pass an anti-lynching law.

Second, you see the roots of the modern Republican party with the alliance between Southern Democrats and Northern Republicans. Four years later, many Southern democrats would bolt from the Democratic party to the Dixiecrats (led by Strom Thurmond) thanks to the inclusion of the civil rights plank. Those dixiecrats would eventually become Republicans (not before supporting George Wallace in 1968) and over the next five decades the Democratic party would lose its historical hold over the South.

Some good general sources: Fear Itself: The New Deal and the Origins of Our Time by Ira Katznelson and Black Ballots: Voting Rights in the South, 1944-1969 by Steven Lawson.