Let $\mathbf{p}_i$ be a finite set of finite-dimensional real vectors with non-negative components with the property that, for any $k$, $\mathbf{p}_k$ cannot be expressed as a linear combination with non-negative coefficients of the remaining vectors $\mathbf{p}_i$.
Consider the set of vectors $\mathbf{x}$ that can be expressed as:
$$\mathbf{x}=\sum_i \alpha_i \mathbf{p}_i \tag{1}$$
for some non-negative coefficients $\alpha_i$.
The question is: Given a vector $\mathbf{x}$, is there an algorithm to determine whether or not there exist non-negative coefficients $\alpha_i$ to express $\mathbf{x}$ as in $(1)$?
Well, this is a small step, but ...
You can first check that the vector $x$ is in the span $S$ of the $p_i$ by computing an orthonormal basis $\{q_j\}$ for $S$ by applying Gram-Schmidt to the $p_i$, tossing out any vector that becomes zero during the GS process. You project $x$ onto this by computing $$ x' = \sum_j \langle x, q_j \rangle q_j. $$
If $x$ is not the same as $x'$, then the answer to your question is "no." On the other hand, if it IS the same as $x'$, then you can express both it and the $p_i$ in terms of the basis $\{q_j\}$. So I've reduced the problem to one in which the $p_i$ span $\mathbb R^n$ and $x$ is a vector in $\mathbb R^n$. I think this is now just a linear programming problem, which you could solve with something like the simplex method, which might take a long time, but it gets the job done. (I'm not at all confident about that last claim, though.)