I am new to Netlify and facing the similar issue when building a Hugo site today.
So I have a private GitHub repo with a public submodule, and allow Netlify to access that repo only.
according to your description, it should check out the submodule automatically during build, but it didn’t.
part of the build log
5:59:40 PM: git clone https://github.com/repo/dummyrepo 5:59:41 PM: Preparing Git Reference refs/heads/master 5:59:42 PM: Found Netlify configuration file netlify.toml in site root 5:59:42 PM: Found Netlify configuration file(s). Overriding site configuration
btw, I do use git submodule add in stead of git clone to add the submodule
path = themes/hugo-ink
url = https://github.com/knadh/hugo-ink.git