I'm aware of the Clean Wehrmacht myth, but I'm just curious if they committed more war crimes than the Allies.
Part 1
Yes.
The Wehrmacht not only quantitatively committed more war crimes but also, and this can not be emphasized enough, the way they waged war during WWII was criminal from the outset as a matter of institutional policy. And not just in the way that the IMT in Nürnberg found that they were guilty of waging a war of aggression but the very way how they fought and what forms of warfare they encouraged was steeped in Nazi ideology.
This started from the very beginning of the war when the attack on Gleiwitz was faked in order to produce a casus belli against Poland, to the regular collaboration of the Wehrmacht with the SS and Police EInsatzgruppen in Poland, to the attack on neutral countries in the Benelux, to the blatant disregard to the laws of war in France where e.g. Rommel both used tactics such as setting civilians' houses on fire in order to mask his troops' advance to feigning surrender in order to draw French troops in, to the anti-Partisan operations in the Balkans, to the war against the Soviet Union being planned as a war of annihilation and thus constituting a war crime in itself, down to the lowest level of the units where massive violence against civilians and their property was encouraged as a tactic of warfare on a massive level.
the Wehrmacht as an institution superseded the "normal" function of an army within your average nation state (this is a bit simplified as neither a normal function or average nation state exists strictly speaking but I mean stuff like defense or fighting a war) and crossed the territory into becoming an institution heavily involved and complicit in the crimes of the Nazi state.
This came to bear in that the Wehrmacht and especially its higher echelons were by the time of the attack on the Soviet Union thoroughly nazified. The war against the Soviets was in their mind not a "normal" war but a war of annihilation. Meaning that civilians as well as the soldiers of the other side were perceived as such an existential thread that extreme violence and terror were the only appropriate measure in dealing with them.
The crimes of the Wehrmacht are numerous. To provide just a couple of examples:
The probably most famous examples of Wehrmacht crimes are probably the Commissars Order and the Kriegsgerichtsbarkeitserlass. When preparing for the invasion of the Soviet Union, the Wehrmacht leadership in conjecture with the Nazi leadership issued orders that the war in the Soviet Union was not to be treated as a "normal" war but a war of "Weltanschauung", meaning they were not just fighting another country but rather Jewish-Bolshevism itself. To that end, the OKW gave the order that political commissars within the Red Army were not to be treated as POWs but were to be shot immediately after capture. Political Comissar included however not only people who held this position but also any member of the Communist Party of the Soviet Union as well as all Jews. In conjecture with this order, the Kriegsgerichtsbarkeitserlass decreed that no member of the Wehrmacht could be persecuted for any and all war crimes they committed while in the Soviet Union. So rape, pillaging, murder and burning down villages were all fair game for all members of the Wehrmacht. The Commissar Order alone lead to something between 60.000 and 140.000 victims.
Additionally, the Wehrmacht as an institution was responsible for Soviet POWs in general. In that function it was the Wehrmacht which basically let them starve to death in violation of all international treaties and conventions. Basically, the Wehrmacht built POW camps for Soviets by just putting up a fence and putting the POWs in there, letting them starve as a policy before the leadership of Nazi Germany needed them for forced labor in 1942. But even with that the death toll is staggering. Christian Streit estimates that about 3.3 million Soviet POWs or 57% of all Soviet POWs captured by the Wehrmacht died while in captivity.
The Wehrmacht was an important part of the occupation of conquered territory and as a security force in that occupied territory. As such, it committed murder and war crimes. Taking Serbia as a territory that was directly administered by the Wehrmacht, Wehrmacht units shot 20.000 civilians alone in the time frame from September to December 1941 as part of a campaign of retaliation for Partisan attacks. The Wehrmacht commander of said territory, Franz Böhme, instituted a policy of 100 civilians shot for every dead German soldier and 50 for every injured German soldier. The vast majority of victims were not related to the attacks or the Partisans but rather male Jews or Roma and Sinti thus making Serbia the first territory outside of the Soviet Union in which Jews were systematically killed by the German occupation.
Crimes such as these are numerous and extend even into the Western territories of Europe. For example, the Wehrmacht massacre of the Italian village of Marzabotto in October 1944.
Also, as an occupational force, the Wehrmacht was responsible for administering Nazi racial policy in its territories as can be read in detail in Dieter Pohl's book on the Wehrmacht in the Soviet Union.
As mentioned above, the Wehrmacht as an institution was involved in the Holocaust in Serbia, where it was Wehrmacht untis who killed the male Jewish population or when it came to Soviet Jewish POWs. But the Wehrmacht also collaborated closely with the Einsatzgruppen in the Soviet Union basically either transferring Jews into the hands of these mobile killing units or even lending a hand when it came to shooting Jews. Additionally, Wehrmacht units in Poland and the Soviet Union also were involved in killing the mentally handicapped and disabled.
Furthermore, the Wehrmacht established Ghettos and provided transport for Jews to be deported to Auschwitz, e.g. in France and aided in registering and confining Jews to certain quarters in countries such as France and Belgium.
The Wehrmacht in fact encouraged its troops to use massive violence against a civilian population as a legitimate means to an end. Dating back to the Franco-Prussian war of 1871, the German army was almost notorious for its fear and hard line against Franc-Tireurs and irregular fighters. The German military doctrine was to employ hard reprisals against the civilian population harboring irregular fighters and security threats. This was allowed by the Hague convention and also used in accordance of it during for example WWI but it was a line of thinking that was quickly adapted to Nazi ideology, meaning that it was applied to Jews ("Where there is the Jew, there is the Partisan, and where there is the Partisan, there is the Jew" as the line from Erich von dem Bach-Zelewski ran) and other civilians perceived as a security threat.
I have mentioned the Serbian example above but this is a general phenomenon in Wehrmacht occupation troops. Rather than punishing violence against civilians, numerous examples prove that the Wehrmacht encouraged violence. Klaus Michael Mallmann for example mentions in one of his articles that a Wehrmacht security unit in Poland was shown the movie "Jud Süß" and became so enraged by it and the copious amounts of alcohol they had consumed that they went out and hunted and shot all the Jews of the town they were stationed in.
Similarly, new research by Sönke Neitzel and Harald Welzer as well as Felix Römer based on newly discovered eve's dropping protocols from British and American POW camps shows that within the Wehrmacht a vast majority of soldiers considered violence against civilians, even women and children in some cases, as a legitimate form of warfare, especially when justified with Partisan warfare. Examples of this, specifically referenced by the Wehrmacht soldiers themselves, include using women and children to clear mine fields; burning down buildings with the inhabitants inside; and the use of public hangings in order to deter support for real or imagined Partisan groups.
The frequency of such happenings as well as the level of involvement on part of the individual soldier are hard to gauge but from all research up to date, it is possible to conclude that almost every unit involved in the war in the Soviet Union or the Balkans did commit atrocities in one form or another on regular basis. Similarly, it is hard to number the victims of Wehrmacht atrocities but even discounting the starved Soviet POWs the number of civilian murdered by the Wehrmacht runs in the several millions.
And that only includes atrocities in the form of murder. The number of rapes committed by the Wehrmacht in the Soviet Union runs somewhere in between 2 and 3 million. This estimate comes from Wendy Jo Gertjejanssen who has her thesis in the subject online here
Also Pascale R. Bos cites a German survey from 1942 in which the Wehrmacht estimates that 750,000 babies had already been born from contact between German soldiers and Russian women. This is a conservative estimate and covers only the time frame up to 1942. While not all of these might have come from rape it shows how endemic the problem was. You can find it in the article "Feminists Interpreting the Politics of Wartime Rape: Berlin, 1945"; Yugoslavia, 1992–1993 Journal of Women in Culture and Society, 2006, vol. 31, no. 4, p.996-1025.