It is possible to enjoy a USA road trip all year round, depending on the region you are travelling to.
For a Western USA road trip for example, peak season is from mid-July to mid-August. That is also when there are the most travellers on the road and prices are at their highest. It is also possible to travel in winter (mid-December to mid-March), but it is important to note that some destinations, such as the Grand Canyon and Bryce Canyon, may even see snow.
Travel in Florida is enjoyable all year round but be aware that hurricane season is from August to October.
To be sure of availability, or to at least make sure that the top attractions are not sold out, you need to book early. For some destinations, such as the American West, many travellers book up to 1 year in advance.
Availability isn't the only reason to book in advance. The earlier you book, the more you save.
So there is really no good reason to wait until the last minute to book your trip to the U.S.!