It appears that what you are looking for is this program – Moyea web player pro -

It is a flash video player maker that provides quick and easy solution to add videos and design a media player on a website. The program will finally publish a html page with the embedded media player with scrollable thumbnail playlist and video files. And this is its tutorial page -
I have used it for a long time to add training videos to our support site and it works well, so hopefully it will help you.

By website builder I take it you mean an html editor

You can go for one like dreamweaver which is really good but will cost you some money

Or you can opt for a free HTML editor, Get a feel of it and then decide if you need something more advanced.
By the way there is a collection of free HTML editors at