The usual proof of Gödel's First Incompleteness Theorem is entirely constructive. We don't have to rely on excluded middle, or have to rely on proving an existential quantification for which we can't produce a witness. For recall: the proof consists in (a) giving a recipe which takes a suitable specification of a sufficiently strong theory $T$ and constructs a certain sentence $G_T$ and then (b) showing $G_T$ is undecidable in that theory. The construction of $G_T$ is clever though simple when you see how, and involves no infinitary ideas. The proof of undecidability involves a pair of reductios, but both of the non-contentious type [like "Suppose $T \vdash G_T$: then contradiction; so $T \nvdash G_T$"]. So overall the proof is intuitionistically acceptable.
The usual proof of the Second Incompleteness Theorem then consists, at heart, in showing that the proof of the First Theorem can be coded up in arithmetic. Again it's all constructive, and so is intuitionistically acceptable.