Let me re-denote your chain complex $a$ by $C$. You can define a chain complex $D$ as the mapping telescope of the infinite sequence $$ \cdots\overset e\to \quad C \quad \overset e\to \quad C \quad \overset e\to \cdots $$ This can be constructed as follows: Form the homotopy coequalizer of the pair of maps $$ 1,S_a: \bigoplus_{\Bbb N} \quad C \quad \to \quad \bigoplus_{\Bbb N} \quad C $$ where $1$ is the identity and $S_a$ is given by applying $a$ and then shifting by one unit to the right in the index. (The homotopy coequalizer is gotten from this diagram by replacing the target $\oplus_{\Bbb N} C$ with its cylinder $\oplus_{\Bbb N} C \otimes I$ and forming the coequalizer of the two inclusions given by $1$ and $S_a$ on each end.)
The effect of this construction is to homotopically invert the map $e$, giving you a model for $C[e^{-1}]$. There is an evident inclusion $i: C \to D$. There is a map $D \to C$ which is defined on the $k$-th summand using the the map $e^{\circ k}$. This will do what you want it to.