This knowledge base article explains how to (1.) acquire a SRT file for a YouTube Video and (2.) convert the SRT file to the WebVTT format.
Although there are a few ways to do this, I found one of the easiest methods is to use the https://www.dvdvideosoft.com/online-youtube-subtitles-download web site. You just need the URL of the YouTube video and voilà, you can download the SRT file.
In a similar easy process we can convert a SRT file to a Web VTT (VTT) format via the https://subtitletools.com/convert-to-vtt-online web site. Just upload your SRT file to the web site and click the button "Convert to Vtt".