I've studied this subject extensively. No, the capitalist system was not in place. It's one of the reasons that Germany was broke. Price controls on most things were strict. Often the cost of production was more than the goods could be sold for. That's a quick way to go bankrupt.
Unions were permitted but the leaders were Nazi party appointees. Most institutions were controlled by the party. And there was a program to take over everything eventually. "Nazi" is the contraction of National Socialist. Hitler knew exactly what he was doing. When the state controls the means of production, the state controls the population and a true dictatorship is the result.
No doubt socialism suited Hitler's agenda for European domination. It nearly worked too. But for his folly in invading Russia, Europe could still be under Germany's control.