Was there any chance of Wilkie managing to win the presidency in 1940?
Yes.
Robert Heinlein discussed the Willkie campaign in his book
Take Back Your Government (written in 1946, after over 12 years of heavy involvement with the California Democratic Party). Heinlein described Willkie as in many ways close to an "ideal candidate": a mature, successful man, physically personable without being "pretty", a good speaker, with an unexceptional family life (as far as anyone knew, including RAH at the time). Also, Willkie was backed by a huge popular movement, ginned up in the space of a few months by some very clever PR and advertising men.
On the other side - Roosevelt was a hugely popular incumbent, but he was running for a third term, which offended a
lot of voters. Many people considered it tantamount to becoming a "president-for-life" dictator.
What defeated Willkie, in Heinlein's opinion, was that he was a first-time candidate who made a lot of unforced errors, saying things that hurt him for no good reason - a syndrome Heiniein had seen before.
If Willkie had better coaching, or learned on his own to avoid these blunders...
I ask this as someone who knows next to nothing about this particular episode in American history. It does seem to me that a Wilkie victory could have had pretty significant consequences domestically and internationally. What would have happened to the New Deal, for instance, or to the United States' relationship to the outside world?
Willkie ran as an opponent of some key elements of the New Deal, notably public-utility operations such as the TVA. His winning would be seen as a mandate to revise or repeal those parts of the New Deal. However, it's almost certain that Democrats would have retained control of Congress - completely certain that they would control the Senate - so Willkie would have trouble getting changes in the New Deal law enacted.
As to foreign affairs: Willkie was privately something of an internationalist. However, he would have won as a Republican, and in 1940 the Republicans were largely isolationist. If the Republicans did really well in the House, they could have a very narrow majority, provided they were supported by the handful of Progressive members.
The Progressive Party had formed as a liberal break-off from the Republicans; most Progressives reverted to the Republican label over time, but a few stayed with the Progressive label. It should be noted that in this era (1910-1940) there were lots of Republicans who might be considered "left-wing", or at least not conservative. For instance, Representative Jeannette Rankin of Montana, was a suffragette and social reformer, the first women elected to the House (in 1916), a pacifist who voted against declaring war on Germany in 1917, was defeated in 1918, returned to the House in 1940, and cast the only vote against declaring war on Japan in 1941. She was also a Republican.
The Progressives were drawn from Republicans of that stripe, who were willing to take up a different party name. And they were fervently isolationist in 1940. If Willkie won... he would be dependent on that Progressive bloc in Congress, and the Progressives by name, if sufficiently offended, might caucus with the Democrats and give them control.
So even though Willkie might harbor internationalist sentiments, he would be constrained from acting on them, much more than Roosevelt..