Let $C'= \bigcup_n nC$ be the cone generated by $C$ and let $X$ be the closed linear span of $C$.
Theorem: A bounded linear functional $q$ with the stated property exists if and only if $C'$ is not dense in $X$.
Proof: Any linear functional which is nonnegative on $C$ will also be nonnegative on $C'$, so if $C'$ is dense in $X$ the only bounded linear functional which is nonnegative on $C$ is the zero functional. Conversely, suppose $C'$ is not dense in $X$ and let $x \in X \setminus \overline{C'}$. Then $\overline{C'}$ is a closed convex set containing the origin and by a standard separation theorem there is a bounded linear functional $q$ and a constant $a < 0$ such that $q(x) < a$ and $q(y) \geq a$ for all $y \in \overline{C'}$. But if $q(y) < 0$ for any $y \in \overline{C'}$ then $q(ny) < a$ for sufficiently large $n$, a contradiction; thus $q$ is nonnegative on $\overline{C'}$. Since $C$ generates $X$ and $q$ is not zero on $x \in X$, it cannot vanish on $C$. ///
Can $C'$ ever be dense in $X$? I think so, even for $p = 2$. Let $(e_n)$ be the standard basis of $l^2$. For each $n$ let $\alpha_n = 2^{-n}$. Then, for each $n$ let $(x^n_k)$ be a sequence in the unit sphere of ${\rm span}(e_1, \ldots, e_n)$ which hits each element of a dense subset of that sphere infinitely many times. For each $n$ and $k$ let $m = 2^n(2k + 1)$ and define $v^n_k = \alpha_m x^n_k + \alpha_m^2 e_m$. Thus, $v^n_k$ is a slight perturbation of a small multiple of $x^n_k$, and all the perturbations are in different coordinate directions. It seems to me that the closed convex hull $C$ of all the $v^n_k$ satisfies $C \cap -C = \{0\}$, yet the cone $C$ generates has every $x^n_k$ in its closure (it contains $x^n_k + \alpha_m e_m$, and the same vector $x^n_k$ appears infinitely many times in the $n$th sequence, therefore with arbitrarily large $m$) and hence it is dense in $l^2$. Proving the first part might take a little work, though.
