This vanishing is very different from Hilbert's theorem 90. I will only sketch a proof as it is rather standard and it will be good for you to fill in the details.

Your module is a permutation module, which means that it has a Galois invariant basis. This is a special kind of induced representation, and the cohomology of induced representations can be computed using Shapiro's lemma. Let $H \subset G$ be the normal subgroup given by the kernel of the representation (note that this is exactly the decomposition group at $\mathfrak{p}$). The module $I_\mathfrak{p}$ might not be irreducible (e.g. if $\mathfrak{p}$ splits), but we can reduce to this case as cohomology respects direct sums. Hence on applying Shapiro's lemma it suffices to show that
$$H^1(H,\mathbb{Z}) = 0,$$
where $H$ acts trivially on $\mathbb{Z}$. However we have
$$H^1(H,\mathbb{Z}) = \mathrm{Hom}(H,\mathbb{Z}) = 0,$$
as $H$ is finite. This completes the proof.