Hello! Short answer is yes, the Wehrmacht was most certainly a mechanized army. They had tank divisions (Panzer Corps) and motorized units (think motorcycles). On top of those there were also troop carrying vehicles.
However, it is important to understand that there were also 3-4 million German soldiers on the eastern front (depending on the year there were more or less). This of course means that the large bulk of those soldiers were not able to be transported through Poland, Ukraine, the Baltic countries, and Russia, and therefore had to travel by foot. This means that many men would either take horses from the local populace for transport, because, as you pointed out, walking to Russia is quite the feat. That is the quick answer, and of course there are many more factors to consider, such as the importance of taking Stalingrad/Leningrad, and attempting to beat the Soviet Union before the Russian winter; but that is a much longer answer for your quick question.
I hope that helps!