If we disregard the Dutch Republic, the Swiss lands and the republican Italian city states, like Genoa and concentrate on the core of the Empire, then yes, even there were a few area that might be called republics.

The most obvious would be the Imperial Free Cities. Each city had it's own constitution, but they were all governed by city council's and mayors (burgomasters). In some cases the guilds might sent representatives to the city council allowing for political influence from both the upper and the middle classes. To be an elected official you had to own significant assents first. As such, the urban poor were always excluded from governance. In general Imperial cities were run along the same way as territorial cities, except they were independent (in theory) from neighboring lords and be allowed to send representatives to the Imperial Diet (Reichstag)

Other areas without lordly control were peasant communities in various peripheral areas. The Black forest was home to the Imperial valley of Harmersbach, which was ruled by independent farmers. It used to be part of imperial cities of Zell, but was pawned away. When the Emperor bought the area back as owner of both city and valley, the council of Zell claimde back the ownership of the area. The local peasants petitioned the emperor for self government and eventually won their case. From 1718 onwards, the Harmersbach would be the only recognised Imperial Valley. In practice, it was governed by the heads of the household of the wealthiest farmers. They were not granted a seat at the Imperial Diet however.

Other rural regions like the Leutkircher heath and scattered imperial villages were likewise ruled by peasant communities. On the North Sea Frisian communities like Hadeln and Ditchmarchen were also ruled without a noble aristocracy.

All this didn't mean these area were governed according to republican ideals of freedom and equality. The idea of the Res Publica was seen not as an a specific system of government. Instead it related to the orderly ruling of the political body, where liberties were safeguarded by the orderly government and the rule of law.