Variations on this question have been asked as far back as Thucydides where he said that if Sparta suddenly ceased to exist, later people would be hard-pressed to believe it was as powerful as people said because of the lack of public buildings, while if the same thing happened to Athens, people would over-estimate its power (paraphrased from Thucydides 1.10).

In the hypothetical you propose, there are several ways we might know about the civilization. First, there would still be archaeological remains, with the survival ranging from really quite a lot if the conditions were right (dry/cold/etc) to smaller amounts in wetter areas. But the key is that there would be metal, stone tools, spaces for wooden construction or even ancient wooden structures like boats, and human remains. Sometimes there is even leather. If you were lucky, they could have preserved their own history orally and had that committed to writing at a time and place that allowed its survival.

Second, depending on where and when this hypothetical civilization existed, other civilizations could write about them leading to other sorts of information. Most likely, there would be some combination of both. Take the Scythians, a group of nomadic pastoralists from north of the Black Sea. Herodotus, a Greek historian wrote extensively about their customs (4.1–4.82), and there have been numerous archaeological finds that give insight into their society (e.g. Parzinger 2016 and stories in Archaeology magazine).

But the hypothetical you pose here isn't simply an issue of "a large and powerful civilization" without stone masonry, but rather something that crops up all the time in looking at the history of people even in settled societies, particularly lower strata of society, let alone during periods of widespread population movements. Fortunately, the answer is yes.

In addition to the great answer you already got, I'd like to add that not all monumental architecture is built of stone. In the Americas, there were powerful societies that built huge earthen mounds. The two major ones that come to mind are the urban cultures of the Mississippi River Valley and the Amazon Rainforest. Cahokia, the biggest Mississippian city, was home to so many people that at its height it may have had more people than London at the same time. There were about 120 mounds there, upon which houses, temples, and other buildings were constructed. Although the buildings don't survive, many of the mounds do, and archaeologists can find evidence of postholes that show where wooden posts supported the structure. In addition to the sorts of artifacts /u/jpnudell covered in their reply, archaeologists can therefore find evidence that there were once substantial buildings, plazas, and other signs of large and stratified populations.

The urban peoples of the Amazon Rainforest also constructed buildings on top of earthen mounds. In the past 20 years or so, archaeologists have found geoglyphs all over the rainforest indicating where large platform mounds once stood. In the case of the Amazonians, there are written accounts which also help fill out this picture. The Discovery of the Amazon According to the Account of Friar Gaspar de Carvajal and Other Documents is a mid-16th century account of a Spanish friar who travelled along the Amazon River with conquistadors who had gotten lost and were trying to regroup with other Spaniards on the Atlantic Coast. Carvajal wrote of the giant cities and roads he saw everywhere they went. For a long time most people didn't believe him because disease had drastically changed the societal structures of the Amazon by the time Europeans penetrated the forest again. But the recent archaeological finds have demonstrated that his account has a lot of merit.

In short, archaeological remains of earthen mounds are another way we know that societies who didn't leave behind stone architecture existed.