The easiest way (I know) to see that there are no nonconstant holomorphic maps from a complete elliptic curve $E$ to the stack $M_g$ is to observe that such a map $f$ would lift to a holomorphic map of the universal covers $\tilde{f}: {\mathbb C} \to T_g$, where $T_g$ is the Teichmuller space. The latter is a bounded domain in ${\mathbb C}^{3g-3}$, so Liouville's theorem implies that $f$ is constant. 

Edit: Using Kodaira's construction of complete curves in moduli spaces (via ramified coverings of products of curves) one can construct maps from elliptic curves $E$ to the coarse moduli space (of large genus) which are generically 1-1, i.e. 1-1 away from a finite subset of $E$. With more work one can probably get injective maps as well but I do not see sufficient motivation for this.