This is not true in general. I don't even see how there would be a natural map between them. There are certainly maps $$ M\otimes N \to M^\vee\otimes N^\vee $$ and $$ M\otimes N \to (M\otimes N)^\vee. $$
The main problem is that tensor product can create torsion and co-torsion and reflexive modules have neither. (The dual of a finitely generated module is reflexive, that is, isomorphic to its own double dual).
Even if you assume that $M$ and $N$ are both reflexive over an integral domain, $M\otimes N$ may have torsion and this gives a counterexample since reflexive modules are duals. Let $M'=M^\vee$ and $N'=N^\vee$. Then $(M')^\vee\otimes (N')^\vee=M\otimes N$ has torsion while $((M')\otimes (N'))^\vee$ being a dual does not.