The problem is equivalent to checking whether the vector $(h_1,\dots,h_m)$ belong to the integer lattice $$\{ Ax \mid x\in \mathbb{Z}^n \}$$ where $A$ is a given $m\times n$ integer matrix. This problem is known to belong to $P$.

However, there exists a similar problem that is indeed $NP$-complete - namely, checking whether a given vector belongs to the integer cone $$\{ Ax \mid x\in \mathbb{Z}_+^n \}.$$

The crucial difference is that in the first problem variables $x_1,\dots,x_n$ can be arbitrary integers, while in the second problem they have to be nonnegative integers. And this nonnegativity requirement turns a polynomial-time problem into an $NP$-complete one.