Are there upper bounds on the L function $|L(E,s)|$ for $|s|<C$?

Assuming the modularity theorem, apply the maximum modulus principle to $$\Lambda(E,s)=N^{s/2}(2\pi)^{-s}\Gamma(s)L(E,s)$$ which is entire and $\Lambda(E,s)=\pm \Lambda(E,2-s)$ (where $N$ is the conductor).

The Hasse bound gives a bound for $\log L(E,s)$ on $\Re(s)=C+2$, this gives a bound for $\Lambda(E,s)$ on $\Re(s)=C+2$ and $\Re(s)=-C$, and since it is entire and rapidly decreasing as $|\Im(s)|\to \infty$ this gives a bound for $\Lambda(E,s)$ on $\Re(s)\in [-C,2+C]$ depending only on $N$ and $C$.

Given a sequence of elliptic curves $E_j$ with conductor $N_j \to \infty$ then $|L(E_j,-1/2)|\to \infty$.