I'm unaware of their being any major study on the matter of PTSD among the German army, let alone the Nazi wing of the armed forces. That isn't to say it didn't exist, but that rather it was exceedingly difficult to study such things in the immediate post war period. However we do know that many German soldiers, both Wehrmacht and SS, both male and female suffered psychological damage during the war. Although its hard to really come up with a steady number or even a rough percentage.

First lets start with the Einsatzgruppen, they were the groups responsible for killing Jews and other undesirables in foreign lands, usually through mass shootings. The members of the Einsatzgruppen were generally volunteers but that didn't change the fact that many had trouble pulling the trigger. Sure there were those members who had no such trouble pulling the trigger (usually Hitler youth members, or other similarly indoctrinated men who were conditioned for these types of activities), but there was also a significant amount of Einsatzgruppen who either refused to kill or needed to be liquored up or shamed into doing the killings. Generally the inability of many Einsatzgruppen to "pull the trigger" was one of the major factors behind the decision to move to mass extermination by gas chamber (the other major factor being that using bullets was considered inefficient). There is an anecdote about what happened when Himmler went to see his one and only mass shooting:

Himmler became very uncomfortable, very quickly. As the firing squad started, Himmler, was even more nervous. During every volley he looked to theground. When two women could not die, Himmler yelled to the police sergeant not to torture them

The same source also says that the leader of the local Einsatzgruppen (Eirc Von Dem Bach) group was angry at Himmler and said this:

Look at the eyes of the men of this Kommando , how deeply shaken they are. These men are finished for the rest of their lives.What kind of followers are we training here? Either neurotics or savages!

There isn't any research that tells whether members of the Eisatzgruppen had higher rates of PTSD than other combat units, but it is generally accepted that preforming atrocities, no matter how well conditioned one is, leads to higher rates of PTSD.

Among the Wehrmacht PTSD was also not unheard of. The army had a name for soldiers who suffered from debilitating PTSD, they were called "Kriegsneurotiker" (war neurotics). Generally throughout the war those who suffered from PTSD either "recovered" and were sent back to the front or if they didn't recover they were classified with the above term. In general, but especially in the lead up to big campaigns like Barbarossa self-mutilations and suicides went up, by the winter of 1941 the number of suicides had already exceeded the amount of suicides in the entirety of World War One (only counting German suicides). After major battles there was also a spike in suicides and self-mutilation, a key example being the aftermath of Stalingrad where there was a huge spike. By 1944 the Army reported that there were over 20,000 "Kriegsneurotiker" and military hospitals became clogged with soldiers suffering mental issues because of the harsh warfare on the Eastern front.

Its harder to pin down numbers for the Waffen-SS and the Gestapo. The Waffen-SS no doubt suppressed cases of PTSD. We do have many diary entries from various Waffen-SS soldiers (I could post some if there is interest) that generally show that while the soldiers may have been frightened, they generally stayed committed and loyal to the cause and to the Nazi party. This isn't to say the Waffen-SS was immune to PTSD but rather that their indoctrination and ideological fervor make it hard to get an accurate understanding of the what the Waffen-SS soldiers really thought.

Sources:

Life After Death Edited by Richard Bessel and Dirk Schumann

Hitler's Second Army: The Waffen SS by Edmund Blandford

Both those books are good for in depth looks into the mind of the average German soldier and the average SS soldier.

The American psychiatrist Robert Jay Lifton was assigned to monitor the psychological makeup of the accused at the Nuremberg War Crimes Trials, including the "doctors trial" where Nazi doctors involved in the "euthanasia" programme, the concentration camps, etc were prosecuted.

He talked to an unnamed Nazi neuropsychiatrist who had treated many members of the Einsatzgruppen, the special units tasked with mass shootings of Jews and communist officials in the Soviet Union. This doctor mentioned that they had the same symptoms as some combat troops: "severe anxiety, nightmares, tremors, and numerous bodily complaints" [all symptoms of PTSD], but that for the Einsatzgruppen "the symptoms tended to last longer and to be more severe". Half of the Einsatzgruppen members that talked to the Nazi psychiatrist attributed their symptoms to the "unpleasantness" of the shootings, and the other half to feelings of guilt. The Nazi doctor estimated that 20% of the Einsatzgruppen suffered from such symptoms. Interestingly, that is about the same percentage of men who at some point refused to carry out (further) executions in Reserve Police Batallion 101, tasked with mass shootings in Poland, and later interviewed by Christopher Browning for his book Ordinary Men.

It should be noted that men who were unable to continue with the killings were relieved from duty and often sent back to Germany for treatment. The goal of the treatment was to relieve the symptoms and get them functioning again so that they could resume their duties. According to the Nazi doctor that had been standard military psychiatric practice in Germany since World War I.

Sources:

Lifton, Robert Jay. The Nazi doctors: Medical killing and the psychology of genocide. Basic Books, 2000.

Browning, Christopher R. Ordinary men: Reserve Police Battalion 101 and the final solution in Poland. Vol. 1998. New York: HarperCollins, 1992.