The question you asked is indeed very interesting. Only in the last few years the history of the “Kriegspfarrer” (war pastors) was researched and there is still a lot to do.
To start with your first question:
There were chaplains in the Wehrmacht and even the SS.
In 1933 the Holy See and the German Reich negotiated a treaty: the Reichskonkordat. This wasn't special at all: after the Lateran Treaty of 1929 the Vatican had to reorganise its foreign affairs. But: the Reichskonkordat had an additional secret protocol added when it was signed in September of 1933. In this secret protocol the Catholic clergy was exempt from future universal army call-ups. Also all students were exempt of military service. (This had to be done in secrecy because Germany wasn't allowed an universal draft due to the Versailles Treaty.)
In case of mobilization the ordained clergy would be part of the Wehrmacht and would be (under the jurisdiction of the church) responsible for pastoral care of the troops. The unordained and members of religious orders would have to work in medical services of the Wehrmacht.
At the start of the war many priests and protestant pastors joined the Wehrmacht, as of February 1940 the Kriegspfarrer were draft. Most priests and pastors did this willingly, were nationalistic and many believed in the German cause. The churches and the state rejected Kriegspfarrer whos political orientation was questionable. There was also the hope to rechristianise the Soviet — and they were suprised that most Russian people still were devout Christians.
The Kriegspfarrer ministered nondenominational services – which many of them didn't like because it was clear that the Wehrmacht's leadership only wanted a kind of diffuse Christianity (and after the Endsieg the end of the churches). But the troops liked those services — especially for the holidays.
Some soldiers wrote home to their pastors. One account I find really astonishing is from the western front somewhere near the English channel: each week, a soldiers writes, they meet up for a 2-hour bible study group, 10 to 15 men, soldiers and officers, even two SS-men.
But most of the SS wasn't in “need” of chaplains — because they were troops that had nearly no connection to traditional Christianity.
So yes, there were chaplains, and for the second part of your question: it was sometimes difficult.
As early as 1941 Josef Reuß who became auxiliary bishop in Mainz after the war knew about the Shoa. The Kriegspfarrer heard confessions and saw the atrocities with their own eyes. Many stayed with the troops because the believed in what they were doing, others stayed for the soldiers' sake.
If there was a theological doubt it was often superseded by the fatalistic faith in God's will and in fate.
In Germany after the war the Churches started to reflect on their own role during the Third Reich. One first step was the “Stuttgart Declaration of Guilt” issued in October 1945 by the Council of the Protestant churches in Germany. It reads (translation from wikipedia):
Through us infinite wrong was brought over many peoples and countries. That which we often testified to in our communities, we express now in the name of the whole church: We did fight for long years in the name of Jesus Christ against the mentality that found its awful expression in the National Socialist regime of violence; but we accuse ourselves for not standing to our beliefs more courageously, for not praying more faithfully, for not believing more joyously, and for not loving more ardently.
There is no mention of the cruelties, no acceptance of a collective guilt.
It is still quite a lot of research to be done to this day, Dagmar Pöppings research is closing this gap of knowledge in the last years.
Literature: