PLEASE READ: To register, turn off your VPN (iPhone users- disable iCloud); you can re-enable the VPN after registration. You must maintain an active email address on your account: disposable email addresses cannot be used to register.
SubRip files (.srt) are certainly an option if available, but I can't say I've ever seen them offered on a website
The captioning on TV broadcasts is actually encoded into an invisible section of the video signal. The information is decoded by your TV and displayed as text. There's no actual voice recognition or anything happening in your TV. It's all typed in by a human being at some point.
Closed captioning is not provided in most (or any!) internet video content. Most formats don't even have the capability to embed captioning data. Quicktime does, but I'm not sure about other formats.
Because internet video is typically lower resolution than TV/DVD, heavily compressed, and also has extraneous video lines removed (the part where the captioning data is embedded) there is no way to extract any existing caption data from web video.
Existing captioning on DVDs can be embedded into Quicktime conversions using HandBrake, a great DVD conversion tool. The captions are even viewable on iPhone and iPod Touch. Handbrake is available for all the main current OS's, free!
