I don't think there's any real way to answer this question with any scientific rigor, at least not as it's posed here. There isn't any statistical data that would allow you to determine whether the "majority" of German soldiers committed war crimes or not. It's not like someone could go down the list of every soldier who served on the Eastern Front and check "yes" or "no" under the "committed war crimes?" tab.

That said, based on what we know about German war crimes in the east, we can say that German personnel from every organization (Wehrmacht, SS, Sipo, SD, Gestapo, Orpo, GFP, FG, etc.) participated in war crimes and crimes against humanity, and that all levels of the Wehrmacht hierarchy, from the OKW down to the lowest-ranking soldiers were involved, as were military personnel from other Axis countries and local collaborators. I'm going to assume that your question refers primarily to the Wehrmacht, rather than the SS and police, so I'll focus on that aspect of it.

Even while the invasion of the Soviet Union was still in the planning phase, Hitler explicitly framed it as a war of ideologies (Weltanschauungskrieg) and a war of racial extermination (Vernichtungskrieg). There was an understanding right away that this would not be a campaign that was conducted according to the standards of international law that existed at the time. The unique nature of the coming war was demonstrated clearly in the criminal orders that were issued by the OKW in the weeks leading up to the invasion: the Barbarossa decree, issued on 13 May 1941, and the Guidelines for the Conduct of the Troops, issued on 19 May, authorized Wehrmacht officers to execute enemy civilians and soldiers who were accused of partisan activity or sabotage, and to apply the principle of collective responsibility if individual perpetrators could not be identified; it should be noted that the term "partisans" included not only actual Soviet partisans, but also implicitly referred to Jews. The Commissar Order, issued on 6 June, commanded Wehrmacht personnel to execute captured Soviet political commissars "immediately, on the battlefield".

The mistreatment of Soviet POWs is one of the better-known examples of the Wehrmacht's war crimes and the revelation of the extent of the mass murder of POWs by German historians was really the key to breaking down the "myth of the clean Wehrmacht", which claimed that the SS was exclusively responsible for German war crimes, while the Wehrmacht was fighting an apolitical war and abiding by international law. Of the 5.7 Soviet POWs captured by the Germans, about 3.3 million died in captivity (about 58%). The collection and transit camps near the front (Armee-Gefangenen-Sammelstelle and Durchgangslager) were often just fenced in open fields, subjecting the prisoners to the elements with little protection. They were generally marched to the main POW camps (Stalags and Oflags) on foot or transported in open rail cars, leading to further deaths from exposure and exhaustion. In the camps themselves, the prisoners received little food or medical care and did not have adequate clothing or housing. In the fall and winter of 1941, the death rate for Soviet prisoners in German captivity reached 1% per day, and it's estimated that more than 2 million prisoners died in this timespan, mainly from starvation and diseases like typhus and dysentery. Mass killing on this scale obviously implicates enormous numbers of Wehrmacht personnel, from the generals at the OKW who ordered the mistreatment of the prisoners to the officers who ran the prisoner of war camps to the rank-and-file soldiers who guarded the camps. Obviously, this depends on how narrowly you define complicity in war crimes, but I think it's relatively uncontroversial to say that even if they weren't pulling the trigger themselves, the soldiers who guarded the prisoner of war camps where dozens of people were dying every day bear some degree of responsibility for those deaths. I don't think there are any comprehensive statistics that would allow a historian to say exactly how many German personnel worked at these camps, but it's safe to say that there were thousands of officers and soldiers who played at least some role in the Wehrmacht's crimes against Soviet POWs.

The Wehrmacht was also extensively involved in crimes against the civilian populations in the occupied territories, particularly in the Soviet Union. While the extermination of Jews and Roma during the so-called "Holocaust by bullets" was primarily carried out by SS personnel from the Einsatzgruppen and local collaborators, some Wehrmacht personnel were also employed as guards and triggermen during the mass shooting operations. Wehrmacht personnel also carried out executions of alleged partisans under the aforementioned orders, as well as reprisals against civilian populations in areas of partisan activity. Wehrmacht personnel also committed some less-known crimes, such as the execution of patients from psychiatric institutions in the Soviet Union (this is separate from the T4 "euthanasia" program in Germany and occupied Poland, which was carried out by the SS).

In addition, the Wehrmacht operated a number of its own camps for civilians. These were mainly improvised facilities that held civilians evacuated from areas near the front lines; they were sometimes referred to as "concentration camps" (Konzentrationslager), but they weren't part of the concentration camp system run by the SS-WVHA. These camps were generally very primitive, often little more than an open field surrounded by a barbed wire fence in which thousands or tens of thousands of civilians were held under inhumane conditions. Death rates in these camps were very high, principally due to starvation, exposure, and disease. It's hard to say how many people died in these camps (or even how many of them there were) because the records are very sparse due to the ad hoc nature of the sites, but based on the research we've done, it's likely that the number is probably over 100,000. Civilians were also conscripted into forced labor battalions under Wehrmacht control, and thousands of women were coerced into brothels that were operated by and for Wehrmacht personnel. Again, it's hard to put a number on how many personnel were involved in these crimes, but these things occurred at many locations on the Eastern Front, and it's likely they were more extensive than is currently documented.

I realize that I haven't really directly answered your question, but as I said at the start, I don't think it's possible to answer that precisely and I don't want to engage in guesswork. However, hopefully this at least gives you an idea of the extent the Wehrmacht's war crimes and the depth and breadth of the personnel who were involved. Even without precise numbers, it's certainly true that a large number of Wehrmacht personnel committed war crimes or were at least present during the commission of war crimes.

Sources:

The United States Holocaust Memorial Museum Encyclopedia of Camps and Ghettos, Volume IV: Camps and Other Detention Facilities under the German Armed Forces (Indiana UP, 2022)

Hannes Heer and Christian Streit, Vernichtungskrieg im Osten (VSA, 2020)

Christian Streit, Keine Kameraden: Die Wehrmacht und die sowjetischen Kriegsgefangenen (JHW Dietz, 1997)

Wolfram Wette, The Wehrmacht: History, Myth, Reality (Harvard UP, 2007)