Smarten Quotes is a Visual Studio Code extension that will convert straight quotes to smart quotes. It works well in text files, Markdown, or HTML. It won't convert quotes in HTML tags or code blocks. It won't convert quotes in Markdown code blocks or tags.
Select the text you want to convert, then open the Command Palette (Ctrl+Shift+P) and select "Smarten Quotes".
No settings.
No known issues. Please report any issues on the GitHub repository.