I have answered this question before, but I am going to update that answer with further information. I'll start with the short answer.

TL;DR: It was widely known both before and during Franklin Roosevelt's presidency that he had suffered some sort of paralytic illness. The extent, however, was probably not widely known by the general public at the time, because there were efforts by FDR's allies to spin it as though he only had some mild, lingering effects of the illness and had mostly got better. The press did not pursue the issue much, out of fear of losing media access to the Roosevelt administration which provided far more important issues of the day. Though the American public was certainly "able to find out" that FDR used a wheelchair to some degree, because of the spin, it wasn't until after his death that the extent of his health issues was generally understood. This started with an authorized biography published in 1954, and culminated with Eleanor Roosevelt's 1961 autobiography, when she wrote at some length about FDR's paralysis. However, there is some academic debate over the extent of the public knowledge at the time of FDR's presidency.

The long answer:

Contrary to popular belief, Roosevelt's battle with "infantile paralysis" was well-known at the time he first ran for the presidency and thereafter. But there was quite a lot of confusion among the public over the extent of the issue, and it wasn't uncommon for people to be under the impression that he had completely overcome the paralysis. That's because there was a lot of contradictory information being reported in the press, which was deliberate. The Democrats were afraid the Republicans would use FDR's paralysis as a political attack, since the Republicans had a history of doing just that. To combat this, the Democrats did everything they could to keep it out of the public eye, and didn't offer anything in the way of details throughout FDR's presidency. For his part, FDR actually did talk about it openly earlier on, before he took office as President, but as one press report put it during his 1936 re-election campaign, Roosevelt "disliked perhaps more than anything else proffered sympathy in his disability". As time went on, there was also fear among some Republicans that any kind of attention to the issue would result in a backlash against them for belittling FDR unjustifiably, and FDR would benefit from a "sympathy vote" so both sides tended to ignore the issue.

Back in 1920, before FDR had contracted his paralytic illness, he had been the Vice-Presidential candidate on the Democratic ticket under James M. Cox, an election which the Democrats lost in a landslide. Though he had served eight years as Assistant Secretary of the Navy, this was the true beginning of him being a national figure. But then the following year, on August 10, 1921, he was struck with the paralytic illness that threatened to end not only his career, but his life. There was a vague press report at the time about him catching a cold and coming down with pneumonia, but it wasn't until September that the press reported on the true extent of the issue when FDR was transported back to New York City. The New York Times broke the story on September 16, 1921, in a front page article under the headline "F. D. Roosevelt Ill of Poliomyelitis". The first paragraph began:

"Franklin D. Roosevelt, former Assistant Secretary of the Navy and Democratic candidate for Vice-President in the last election, was brought to this city from Campobello Island, Bay of Fundy, yesterday, suffering from poliomyelitis, or infantile paralysis, which for more than a month has caused the loss of the use of both legs below the knees."

In truth, both his entire legs were affected, not just below the knee. He had earlier on experienced temporary paralysis in his upper extremities as well though that had since subsided.

Over the next several years, there were occasional stories about his recovery, notably one published in the Sunday Magazine of the Atlanta Journal on October 26, 1924, that was syndicated throughout the country. It detailed FDR's patronage of the Warm Springs hydrotherapy facility in Georgia, and was one in a long line of articles that suggested FDR was getting better:

"Mr. Roosevelt does not know how he contracted the dread disease, and does not regard himself as more outstanding or unfortunate than the hundreds of other adults who became victims at the same time by the disease usually confined to childhood. All he does know is that he was hit, and hit hard, with the result that both of his legs were immovable for many months. Gradually he acquired the skill necessary to drag himself around on crutches, and, undaunted, he was a prominent figure in the Democratic national convention in New York last June, making a memorable address..."

An article about FDR's involvement with Warm Springs, published in the August 13, 1927, edition of the New York Times, foretold the contradictory nature of how the issue would be reported in the press in the future. One of the sub-headlines in the article indicated FDR had completely overcome the paralysis: FDR "Has Dispensed With Crutches". But just two sentences later, the article notes he was progressing, but not quite that far: "In his own case, he said, since taking the treatment [at Warm Springs] he has been able to abandon one leg brace and substitute a stick for crutches."

By then, the political attacks against him on the basis of his health had already begun, so the suggestions of improved health were deliberate political "spin" to minimize these attacks. As mentioned in the aforementioned Atlanta Journal article, he had delivered a speech at the 1924 Democratic National Convention on behalf of New York Governor Al Smith, in hopes of getting him the nomination for president (it didn't happen that year, but Smith did get the nod four years later). Papers around the country had reported on FDR's speech and mobility issues, and often, partisan Republican papers were not kind. The Chicago Tribune was perhaps the most vivid, in an article published on June 27, 1924:

"He is hopelessly an invalid; his legs are paralyzed. Wheel chair, crutches, and attendants are with him wherever he goes."

In 1928, he ran for governor of New York, to replace Al Smith, the Democratic nominee that year for president. It was the first time Roosevelt had run for public office since the onset of his paralytic illness. According to David M. Oshinsky in Polio: An American Story, the Republicans' campaign strategy in the gubernatorial race was to exploit FDR's physical issues while playing up the strong economy under the leadership of Republican President Calvin Coolidge:

"The campaign was short but vigorous. Nominated in early October to oppose Albert Ottinger, the state’s attorney general, Roosevelt spent the remaining four weeks in search of votes. The Republican strategy was simple: play up national prosperity on the one hand and Roosevelt’s disability on the other. According to the GOP, a desperate Al Smith had forced a noble but crippled friend to reenter the political arena against the advice of his doctors—and perhaps at the risk of his life. Republican newspapers had a field day, describing FDR’s campaign as 'pathetic,' 'pitiless,' 'unfair to Mr. Roosevelt,' and 'equally unfair to the people of the State.' Al Smith’s response would become a part of American political folklore. 'A governor does not have to be an acrobat,' he said. 'We do not elect him for his ability to do a double backflip or a handspring. The work of the governorship is brainwork [and] there is no doubt of [Franklin Roosevelt’s] ability to do it.'"

FDR won a very close election in a year that was otherwise rather abysmal for the Democrats. Two years later, in 1930, FDR won re-election as governor by a wide margin, with the Stock Market Crash and the Great Depression largely blamed on Republican leadership. FDR became an early favorite for the Democratic nomination for president in 1932, but re-entering national politics meant dealing with FDR's health issue. Already aware of the way the issue had been used by the Republicans in 1924 and 1928, the Democrats didn't want it to be used against them again. The fear was real. In April 1931, an opponent of FDR's possible candidacy, Jesse W. Nicholson of the National Women's Democratic Law Enforcement League, asserted in a widely reprinted quote:

"This candidate, while mentally qualified for the Presidency, is utterly unfit physically."

cont'd...