Is there an example of a commutative ring $K$ and modules $V_1, V_2, W_1, W_2$ such that the canonical linear map $\operatorname{Hom}(V_1, W_1) \otimes \operatorname{Hom}(V_2, W_2) \to \operatorname{Hom}(V_1 \otimes V_2, W_1 \otimes W_2)$ fails to be injective?
Certainly it is always injective in the case where these are all free, and it is bijective in the case of finitely generated free modules.
Yes, there are many such examples. A neat way to find some is to manage to get $W_1\otimes W_2 = 0$ without having the LHS vanish.
One such example is given by $ W_1=W_2 = \mathbb{Q/Z}$. Then $W_1\otimes W_2 = 0$ so the RHS is $0$, so it suffices to find $V_1,V_2$ such that the LHS is nonzero, and then the map will automatically fail to be injective.
Take, e.g. $V_1=V_2 = \mathbb Z/n$ with $n>0$, then $\hom(V_1,W_1) \cong \mathbb Z/n$ (the $n$-torsion of $\mathbb{Q/Z}$), and so the LHS is $\mathbb Z/n \otimes \mathbb Z/n $ which is always nonzero for $n\geq 2$.