Excluding the Jews, they did. Two of my great uncles in law served with Ukrainian units in the German army. They had the tattoos to prove it.
I believe 50,000 Jews served in the Wehrmacht. Some openly practising their faith. The Nuremberg Laws automatically gave exemption to any Jew that had earned an Iron Cross and his family, any that was only 1/4. Even these requirements could be over ruled by the authorities or Hitler personally. There are complex and nuanced aspects to this. Even the head of Luftwaffe procurement, Erhard Milch had a shadow cast over him because of a Jewish mother and that was over ruled by Goering. Most Germans lived in smaller towns, most German Jews lived in bigger towns or cities. This whole field is used as a modern day political and ideological rhetorical football and we are unlikely to see what ordinary people at the time felt, saw and experienced. It's why its best to stay clear of it.