JotForm is a free online form builder which helps you create online forms without writing a single line of code. No sign-up required.
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You can add an upload button to let a user upload a video of themselves.
Then add the list of common video formats in the Extensions field.
mpeg, mpg, mpe, avi, mov, qt, wmv, rm, mp4, 3gp, mkv, flv
You may also need to increase the upload size limit. Here's a guide that can help: Changing the Upload Size Limit.
Furthermore, please note that you might need to have a paid account, as free accounts only allow a total upload space of 100 MB.
Or if you want users to submit a youtube link, you can simply add a regular textbox field.
Do let us know if you need further assistance.