I don’t know WordPress has its own video player, but HTML5 video player supports closed captions.
This is the example.
<video width="320" height="240">
<source type="video/mp4" src="/my_video_file.mp4" >
<track src="/captions_file.vtt" label="English" kind="captions" srclang="en-us" default >
<track src="/French_captions_file.vtt" label="French" kind="subtitles" srclang="fr" >
</video>
But, I recommend this Video.js