Uhhh not really no?
The word comes from PGmc *rīks as I mentioned. It's a PCeltic borrowing meaning "king". The adjectival form of the root - *rīkijaz "kingliness" refers to any kinglike quality. So it could mean, depending on context: powerful, rich, mighty, royal, noble, etc. This is where our adjectives: reich, rijk, riche, rich, rik, etc. come from.
The neuter substantive form *rīkija refers to "kingliness" in an abstract sense. So rulership, authority, and government when referring to the organization and means of ruling, but additionally kingdom, realm, and later (as PGmc transitioned into languages for which we have actual hard evidence such as Frankish, OHG, OE, and Gothic) empire when referring to things a king rules over. This is where the neuter substantives I listed above: das Reich, het rijk, þat rike, riket, etc. come from. And as I noted, the word is still used in e.g. German and Dutch when describing states English would classify as "Empires" - Mogulreich, Persereich, Mongolisches Reich, Guptareich; Mogolrijk, Perzische Rijk, Mongoolse Rijk, etc.
I think the problem with this discussion is in the nature of the question, because the question and the answer kind of end up becoming tautological. Germanic languages obviously had concepts of kingship and words to describe kingship and the realms over which a king ruled. The concept of an overking or High King also obviously existed. So in that sense an "Empire" as in a "Reich": a large territorial state ruled by an Overking, absent any explicit associations with a Roman heritage was something that the Germanic language had, and still has today. It's also worth noting that, for example, German is a bit more explicit in distinguishing between Empires claiming (whether through themselves or later Western historians) association with Rome. So you get non-Roman "Empires" like Guptareich, and Malireich ruled not by Kaisers, but by Königen, and then you get the more explicitly "Roman" Empires like Deutsches Reich, Russisches Kaiserreich, and Byzantinisches Reich which are ruled by Kaisers. The problem is that "Empire" as a word has a lot of baggage attached to it, which, due to reasons of both etymological and cultural heritage necessarily associate the term with Latin and the Romans. Many western languages adapted the Roman titles Caesar and Augustus specifically because of the associations that word had and has with, well, the Caesars. So this whole discussion kind of turns into a big tautological mess. "Did Germanic speakers have a word specifically to describe the Roman Empire and its leaders as distinct from any other large territorial state with a High King?" No, obviously, but then, why would they? And perhaps more critically, neither did the Romans.