As far as I can tell your question is "can I reconstruct a polynomial from its roots." So, yes, you can consider the product $\prod_{\alpha} (x-\alpha(H))$ as a function on $\mathfrak{h}$ valued in polynomials. This is the same (by the definition of root) as the characteristic polynomial of the adjoint action by $H$.

As far as I can tell your question is "can I reconstruct a polynomial from its roots." So, yes, you can consider the product $\prod_{\alpha} (x-\alpha(H))$ as a function on $\mathfrak{h}$ valued in polynomials. This is the same (by the definition of root) as the characteristic polynomial of the adjoint action by $H$ (on $\mathfrak{g}/\mathfrak{h}$; for all of $\mathfrak{g}$, you should multiply by $x^{\mathrm{dim}(\mathfrak{h})}$).