How to make VS Code treat files without extension as a certain language?

You can apply the file associations to all files in a given directory only:

"files.associations": {
    "**/RootDir/**/*": "COBOL"
}

This way, all files in /RootDir/, or one of it's sub-directories, will be mapped to COBOL.
All files outside of /RootDir/ will still be mapped as usual according to their extensions.

Suppose your /RootDir/ doesn't contain only COBOL-files, but also some other file types. In this case you can go further and define exceptions for the file associations:

"files.associations": {
    "**/RootDir/**/*.bat": "bat",
    "**/RootDir/**/*.sh": "shellscript",
    "**/RootDir/**/*": "COBOL"
}

Basically you're instructing Visual Studio Code to map all files in /RootDir/ to COBOL, except for .bat and .sh which are mapped as Batch and Shellscript, respectively.
As above, all files outside of /RootDir/ will still be mapped as usual according to their extensions.