Yes.
If you require that no difference $p_j-p_k$ is constant (which follows from
your assumption $p(0)=0$), then $c_1e^{p_1}+\ldots+c_me^{p_m}=0$ implies that all $c_j=0$.

In fact a more general result is true: instead of polynomials one can take any
entire functions. This is called Borel's theorem.
(See, for example, S. Lang, Introduction to complex hyperbplic spaces, Springer 1987, Theorem 1
in VII, sect 1, p. 186.)

For polynomials in one variable, this is easy: look at the asymptotics,
for example when $z$ tends to infinity on an appropriate ray, and use induction
in degree of the polynomials. To obtain the same for several variables, restrict your identity onto lines in $C^n$.

means"every finite subset of $X$ is linearly independent". $\endgroup$