Did the Holy Roman Empire as an independent entity have any American colonies? To that, I'd say the answer is no. However, some of the Imperial states held limited colonial possessions. However, even then, we're not talking about large scale colonisation, on the scale of the big colonial nations like France, Spain, and Britain, but rather small settlements and outposts.

Brandenburg-Prussia held some small islands in the Caribbean and Africa, and the Austrian monarchy maintained some similar possessions in the Indian Ocean. None of these were really major developments, but they did exist.

Now, onto the question of why. As for the Holy Roman Empire itself, it's quite simply that the Empire didn't have the institutional support to develop and send out colonisation fleets of the other powers. The Emperor had hard enough trouble getting the Imperial Estates to support military expenditures to fight the French and Turks, and was unlikely to find the political capital to support an 'Imperial' colonisation effort. Moreover, given the position of the HRE, it didn't have the same easy access to the Atlantic as many of the Western European powers. While there were major trade ports in Northern Germany, they're still hemmed in by the various other powers in the Baltic, the North Sea, etc.

An interesting example comes from the Ostend Company. After the War of the Spanish Succession, the Austrian Habsburgs received the Spanish Netherlands, which included major ports like Antwerp and Ostend. The Austrian Habsburgs were, of course, quite interested in using their new possessions to tap into the lucrative East India and China trade. Emperor Charles VI granted a charter to the Ostend East India Company in 1722. At first, the Ostend Company was very successful, and--at its height--was bringing in nearly half of Europe's tea imports, with trading posts in India, much like many other European trade companies.

However, this success was viewed as a threat by many of the other Maritime Powers, including the Dutch and English. As such, the Maritime powers increased pressure on the Austrians in order to revoke the company's charter and dismantle it, eventually Charles VI buckled, and agreed to dismantle the company. Quite simply, the Austrians needed the money and support of the Maritime Powers (England and the Dutch) in order to counterbalance the power of France, and retain support for the Pragmatic Sanction that was--allegedly--to allow Maria Theresa to ascend to the throne (Charles VI being without a male heir).

So, we have a few reasons for why there was no major German state colonisation during the Age of Colonisation. The Empire itself didn't have the unity to sponsor such expeditions, nor did the individual German states. Geographically as well, the German states weren't in position to strike out into the Atlantic like Spain, the Dutch, France, or England. Finally, there was diplomatic pressure: the colonial powers didn't want any additional competition, and were willing to use diplomatic pressure to cut off colonial ventures.

Hope this answers your question, and if you have any follow ups, please feel free to ask!