The Wehrmacht did not exist until the Nazis seized power. Until 1935 - a year after Hitler had already combined the role of Reichskanzler and Reichspresident under the 1934 Referendum - the German military was known as the Reichswehr, and at least on paper it was still bound by the terms of the Treaty of Versailles limiting its size and equipment. In 1935, with the reintroduction of conscription (banned by Versailles), the Reichswehr formally became the Wehrmacht. The Wehrmacht, despite the best efforts of its postwar propagandists to portray it otherwise, was from the roots an instrument of Nazi power. Hitler knew that a loyal military was an absolute necessity for his political goals, and he maneuvered accordingly. This took two major forms: pushing National Socialists to positions of power within the military, and manipulating the motives of non-Nazis to gain their support. A good example: from 1934 onward, soldiers of the Reichswehr/Wehrmacht swore an oath specifically to Hitler, replacing the previous oath to the Weimar Republic. The so-called 'Hitler Oath' was offered up unprompted by then-Minister of Defense Werner von Blomberg as an expression of loyalty. Blomberg was not, as I am aware, a committed National Socialist; however, he was a staunch 'Wehrstaat' conservative militarist, as were many of the Reichswehr leaders, and Nazism was a very attractive movement because it promised rearmament and the assertion of German power on the continent once more.

It is true that the Wehrmacht was not solely occupied by National Socialists during the 1930s. Some, like Ludwig Beck, were perhaps (as Shimon Naveh calls them) 'technocratic opportunists,' conservative militarists who glommed on to Nazism as a useful movement for their secularly militaristic and nationalistic goals. However, from 1939 onwards, NSDAP party membership was a prerequisite for national service (though some exceptions did exist among the senior officer class), and the German officer corps was somewhere between a willing instrument and an active, at times rabid, proponent of the National Socialist project in its entirety, from rearmament to expansionism and reclamation of lost territory to genocide.

Further reading - Richard Evans' Reich trilogy is indispensable, as are Geoff Megargee's Inside Hitler's High Command and, in a more generalized sense, Rob Citino's Wehrmacht trilogy, if you want to understand what made the Wehrmacht tick. Ian Kershaw's Hitler biography also has a lot to say about the relationship between Hitler and his military, as does his 1993 article "Working Towards The Fuhrer" in Contemporary European History.