Have bicycles ever had a large role in the U.S. armed forces?

Bicycles were very in vogue as a military weapon around the turn of the century (19th to 20th, that is). To be frank, I don't know much about American use, but I have written about their role in the military as a whole at that time, which might interest you.

Military bicycles were very much in vogue for a time, starting in the late 19th century, and a number of European armies, by the first decade of the 20th century, had organized entire battalions of cyclists, usually mounted on folding bicycles. The doctrine for them was for the bikes to be folded, and carried on the back with a special strap during battle, which you can see here with French cavalry's bicycle battalion doing drills with their mounts on their backs. Every French Cavalry Division would have a battalion of bikers attached to it, which was common in other armies to at the time. The Belgians only had a single battalion in their army but the bicycle troops were a point of pride and as such the Carabinier-cyclist Battalion was considered to be an elite force within the Belgian Army c. 1914. They were hardly the only ones to have such units though. The Germans and the Italians went bike crazy too. As for combat use, the units certainly saw combat, but if you are hoping for "cavalry charges" on bikes, I think you'll be disappointed, at least insofar as my reading has shown. Bike troops were more akin to dragoons, using their mounts to get to the fight, but dismounting upon arrival.

Bikes remained in use after World War One, as we can see with these Hungarian bicycle troops in the 1930s, and during World War II especially found new life as a lightweight form of transport that paratroopers could bring with them. Here are some Royal Engineers loading their airborne bikes into a Hora Glider. Ground troops still employed them though, perhaps most notable being the Finnish Jaegers, who would trade in their skis for bikes during the warmer months of the year.

Following World War II, I'd say that the use of bikes declines markedly, and bike units were much less of a thing, although a little digging around shows that the Swiss maintained regiments of them until 2003(!) and of course although units organized around them no longer exist, that doesn't mean you don't hear about there occasional use, as, for instance, I recall reading about Special Forces in Afghanistan making use of them.