Actually, looks like emscripten has support for a few different options, including IndexedDB[1]. I have a use case where we'd like to both use the local filesystem via Node APIs as well as in-memory in the browser. I asked the author of wasm-git and it looks like this is possible with a custom build[2].
Would it be a security risk to grant limited local filesystem access by domain; with a storage quota?
... To answer my own question, it looks like the FileSystem API is still experimental and only browser extensions can request access to the actual filesystem: https://developer.mozilla.org/en-US/docs/Web/API/FileSystem