Currently it is really difficult to find out whether there are subtitles for a video, and what's their URL and other data.
action=query&prop=videoinfo&viprop=timedtext&titles=File%3AEdward%20Snowden%20speaks%20about%20NSA%20programmes%20at%20Sam%20Adams%20award%20presentation%20in%20Moscow.webm
should return something like (similar to the "derivatives" data):
"videoinfo": [{
"timedtext": [{
"src": "https://commons.wikimedia.org/w/index.php?title=TimedText:Edward_Snowden_speaks_about_NSA_programmes_at_Sam_Adams_award_presentation_in_Moscow.webm.en.srt&action=raw",
"type": "text/x-srt",
"kind": "subtitles",
"srclang": "en"
}]
}]