Question: Did FDR support Herbert Hoover for President in 1920?
Hoover was a wealthy, admired, and world famous man before WWI. His humanitarian efforts both as a private citizen and later as an agent of the United States increased this fame and admiration by a few orders of magnitude. FDR during WWI was the Assistant Secretary of the Navy, by comparison nearly a nobody. I say nearly because FDR was still tied to Teddy Roosevelt and thus still commanded more fame and name recognition than his achievements up to that point probable merited.
So did FDR support Hoover for the Presidency in 1920. FDR who was supremely political, tried to recruit Hoover for the 1920 Democratic Presidential Ticket, with himself as the Vice President. Unfortunately Hoover's fame was so great that he was also being courted by the Republican Party for their nomination. Hoover who's fame was magnified as a President Wilson(d) political appointee, declared himself a Republican and rejected FDR's proposition. As you mentioned FDR later obtained the vice president spot on the Democratic Ticket from James Cox the former governor of Ohio who secured the Democratic Nomination. FDR was 38 in 1920, and was four years younger than Teddy was when he was nominated for the same office. The Cox Roosevelt democratic ticket was soundly defeated by Warren G. Harding and Calvin Coolidge in the 1920 Presidential Election with the Democrats only taking a few states in the south. Hoover was given the relatively low profile position of Secretary of Commerce in the Harding administration.
Hoover would win the Presidency in the 1928 election taking office March of 1929 a few months before the Great depression began(October 29, 1929). It was his first elected office all his previous government positions were as an appointee.
More Detailed Answer
World War I ended in November of 1918. Franklin Delano Roosevelt(FDR) had been the assistant undersecretary of the navy under President Wilson. Herbert Hoover was a significantly more famous guy in 1920 than was FDR but was not from a political background and was thus somewhat of an unknown political entity.
Herbert Hoover was a famous and very wealthy engineer from the private sector who was famous for challenging engineering projects across asia (Australia and China) before WWI, and became even more famous during and after WWI.
Before the U.S entered WWI, citizen Hoover organized and lead an effort to repatriate 120,000 Americans from Europe. He lead and organized 500 volunteers who went to Europe and gave away food, clothing, cash and steam ship tickets back to the U.S. to Americans trying to return home.
As a private citizen, Hoover then organized and lead an effort to purchase and distribute food to civilians on both sides of the war. Hoover's organization basically feed Belgium for the duration of the war and at its height served 10.5 million people daily, with a budget of 11 million dollars made up from both Government Grants and private donations.
When the United States entered the War President Wilson appointed the famous and respected Hoover to lead the U.S. Food Administration. A kind of food Czar who would stockpile and secure American food supplies(purchase, store, and transport) for the war effort. After the War Hoover's Food Administration was tasked with feeding Central and Eastern Europe until their own economies got back on their feet. Something Hoover achieved with remarkable proficiency.
All this high profile service made Herbert Hoover a popular figure in the United States and abroad. His nickname was the Great Humanitarian. Young 38 year old FDR recognized Hoover's fame could be turned into political currency and tried to recruit him for the head of the Democratic Ticket. It all fell through though when Hoover declared himself a life long Republican and rejected FDR's overtures. Herbert Hoover would accept the relatively minor cabinet position as Secretary of Commerce in the Harding Administration.
After WWII the Truman Administration would again tap Hoover in the same role of organizing US food aid to Europe. In 1945, Hoover would again organize and distribute food across Europe averting mass starvation and do all this millions of dollars bellow what the Truman administration projected for the relief effort.
Wiki: Herbert Hoover
Wiki: Franklin Deleno Roosevelt
NY Times: Herbert Hoover backed for the 1964 Nobel Peace Prize