This is not true in general. There are certainly maps $$ M\otimes N \to (M^\vee)^\vee\otimes (N^\vee)^\vee $$ and $$ M\otimes N \to (M\otimes N)^{\vee\vee}. $$
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.
Notice that in this case there exists a map $$(M')^\vee\otimes (N')^\vee \to (M'\otimes N')^\vee$$ but it is not injective.