Let $E$ a vector bundle on a differentiable manifold and $D: E\rightarrow E\otimes \Omega^1$ an homomorphism, with $\Omega^1$ differential 1-forms. If I take the map $D\wedge D$ which is the target space?
In this context the wedge product can be defined for differential forms, so you will need to see your $D$ as a differential form with coefficients in $E$. For this purpose it should be defined as $D:\Gamma(E)\rightarrow \Gamma(E)\otimes \Omega^1$ with desired properties, where $\Gamma(E)$ denotes the space of sections of the vector bundle $E$. However, if you attempt to define then a wedge product of such sections you will run into a problem how to multiply sections of $E$. The bottom line is that you do not have enough structure yet to define $D\wedge D$.
(This probably should be a comment but I currently do not have enough reputation).