Given an oracle that can solve in polynomial time:
$$a^Tx=b$$ $$x \geq 0$$
So it can solve the feasibility problem with one equality-constraint ($a$ is here a vector and $b$ is a constant, $x$ is required to be integer) and non negativity constraint for the solution. (This problem is also known as unbounded knapsack problem)
With this oracle how can this be solved in polynomial time: $$Ax=b$$ $$x \geq 0$$
where there are arbitrary many equality-constraints ($A$ is here a Matrix and $b$ an vector, $x$ is required to be integer) and non negativity constraint for the solution.
Just finding a solution for every constraint is not enough. You need to somehow be able to find a solution which fulfills all constraints. But i don't see a way? But there has to be way, because both problems are NP-complete.