Yes, this is Theorem 5.5.11 in Spanier's "Algebraic Topology" text.

The conditions are that the torsion product $\operatorname{Tor}_R(M,N)=0$ and either $H(A;M)$ and $H(B;N)$ are of finite type, or $H(B;N)$ is of finite type and $N$ is finitely generated.

Then there is a natural short exact sequence
$$
0 \to H(A;M)\otimes H(B;N)\to H(A\otimes B; M\otimes N)\to Tor_R(H(A;M),H(B;N))\to 0
$$
(where the second map raises degree by one) and this sequence splits.