I'm not sure you need the splitting principle (which isn't mentioned in the book by Milnor and Stasheff): if you prove the formula for canonical bundles $\gamma^m,\gamma^n$ over the grassmanianns $G_m, G_n$, you can use the existence of $\iota_m:\xi^m\to\gamma^m$ and $\iota_n:\zeta^n\to\gamma^n$ to deduce the formula for $\xi^m,\zeta^n$ (no need for $\iota^*$ to be injective in that sense !)

So $w(\gamma^m\otimes\gamma^n)\in H^*(G_m\times G_n)$, which Künneth's theorem computes from $H^*(G_m)$ and $H^*(G_n)$, showing us that $w(\gamma^m\otimes\gamma^n)$ is some polynomial in the Stieffel-Whitney classes of $G_n$ and $G_m$.

To express this polynomial, we split $G_n$ and $G_m$ over the cartesian products of $n$ (respectively $m$) infinite dimensional projective spaces (let us denote by $\iota_n,\iota_m$ the morphisms from these products to the grassmannians) [Milnor, Stasheff] proves that the corresponding $\iota^*$ are injective and that the SW classes of $\iota^*(w_k(\gamma^n))$ is the elementary symmetric polynomial of degree $k$ in $a_1,\ldots,a_n$ where $a_i$ is the SW class of the canonical line bundle in the $i$_th copy of $P^{\infty}$.

This is for our right hand-side in Künneth's theorem. What happens to $w(\gamma^m\otimes\gamma^n)\in H^*(G_m\times G_n)$ when we split? It becomes $w((\gamma^1\times\ldots\times\gamma^1)\otimes(\gamma^1\times\ldots\times\gamma^1))$ with $m$ copies first, $n$ copies then. Just distribute, use the formula for the SW class of a cartesian product and the fact that $w_1(\gamma^1\otimes\gamma^1)=a_i+b_j$ where $a_i,b_j$ are the $w_1$ of each copy of $\gamma^1$ to conclude.