The HRE did not act as a single entity to the outside, for example a country at war with Austria would by no way automatically be at war with the other members.

The Hapsburg monarchy (the emperor) did not offically support the American cause, but conducted trade anyway. The American embassy to Austria gives a nice little summary of the relationship here.

Quote:

"In 1777 Congress appointed William Lee as the first U.S. representative to Vienna. Austria, however, did not receive him, declining to recognize a country that had rebelled against a monarchy. The Hofkanzlei’s files referred to American officials as “rebels” and “insurgents.” The Empire nonetheless conducted trade in arms with the colonies during the American Revolution, and pursued commercial interests in cotton and tobacco markets."

As /u/xacisx has already pointed out, the HRE was not a single entity in international politics. It was conceived as a kind of federal union right from the start and often dominated by internal struggles and conflicts between the dynasties (e.g. Habsburg, Wittelsbach, Luxemburg, etc.) and their territories within the HRE. As the years and centuries progressed, these struggles became more and more frequent and aggressive (think 30 Years War) and the HRE became even more fractured while the power of the Emperor (which had never been as great as some people might think) dwindled. By the time the USA came into existence, the HRE had been dominated by the Habsburgs for a while, who started focussing more and more on their own territories (which included lands outside the HRE, i.e. Hungary), thus making common foreign policy impossible for the HRE.