You can find the detailed proof in my joint paper with Olexandra Khokhliyuk for the case of weak Whitney topology:
 O. Khokhliuk, S. Maksymenko,
Smooth approximations and their applications to homotopy types.
Proceedings of the International Geometry Center, vol. 13, no. 2 (2020) 68-108
I needed this kind of results two years ago and searched for the proof but failed to find it.
Finally we decided to write the detailed proof.
The paper also contains variants for manifolds with corners, inclusions of open sets (as mentioned by Ryan Budney here), and relative variant for maps with fixed restriction to some subset.
It also contains references to partial results and some posts in the internet including the present post by Adrian Clough.
When we put the preliminary version of the paper to arXiv, Helge Glockner informed us that such a statement follows from his (currently yet unpublished) paper
 Helge Glockner.
Homotopy groups of ascending unions of infinite-dimensional manifolds (2008)
Then we also included (in Remark 7.5 of ) explanations of how to get the mentioned result from Helge Glockner's paper .