Hi there, I'm Matt Sergej and believe I can implement your web application (if I'm free next 30 days, will know on Wednesday afternoon) with 24 years of IT development, web mastering and web animations/visualizations (CSS3, SVG, javascript) experience.
Otherwise I understand perfectly what you need - have two live courses on Udemy where they have upgraded their captions from SRT to VTT in the last year if I recall. And know YouTube subtitling process with more experience of course with Camtasia Studio and Aegis.
Looking forward to receive any additional application requirements. PHP can suit the tasks, the missing information for me from your description is about saving the VTT files (what do you plan on the webserver for that).
The deliverable will be a web application with web page, javascripts, CSS styles and all necessary PHP code for interacting with the web server (saving VTT files to a database, authentication is not part of this bid, assuming you have it - can help here though too).
Best Regards,
Matt