Vienna sent a trade delegate to the US directly after the signing the Treaty of Paris at 1793. Albeit no arrangements were made amongst Empire and US since Emperor Joseph II did not recognize the US. In this manner there was no formal connection amongst the Empire and US until 1802, when Emperor Franz II endorsed a US consulate in Trieste. This implies that at least Franz II, as the HRE emperor at that time, recognizes the US.
So suppose HRE, as a diplomatic entity, finished in 1804 when Franz II changed his title from HRE emperor to the emperor of his recently established Austrian Empire. In fact there should in any case be a multi year time span somewhere in the range of 1802 and 1804 when US and HRE formally recognize each other. Edit : fixed typo
/u/George_s_patton_III
Rather than directly answering your question, to which the answer is NO, I will point out that you seem to be assuming that in 1776, both the US and the HRE were sovereign nations. And thus, it should be reasonable whether two sovereign nations had diplomatic contact. The real issue here is that the HRE was not a sovereign nation as much as it was a system of politics, privileges, and commitments.
To quote a prior post,
In Westphalia, that nations such as France and Spain were sovereign was not new. What was new was that small nations were sovereign too, and had their own voice. This was very important in the context of the member states of the HRE.
Further, as I have posted here, varying degrees of sovereignty, in the sense that member states could conduct their own foreign policy, had started in the late 1400s and continued until it was formalized in 1644 during the Thirty Years' War.
You could then ask, did any member of the HRE have diplomatic contact with the US. The answer is elsewhere in this thread.
You might be interested in this answer by u/Erusian to a similar question a few months back. https://www.reddit.com/r/AskHistorians/comments/7tnb1z/there_was_a_23_year_period_between_the_peace_of/dtfiox3/
I've nothing to add about the Hapsburgs, but you may be interested in an article I wrote on the proposed alliance between the United States and the Knights of St John of Malta, which you can read in toto here for the Journal of the American Revolution.