You need an article for "website", otherwise it's unclear what you mean. By "headhunter" I assume you mean a proactive employer. "The headhunters came to you on their own through a website." = independently, without you having to approach them. They contacted you "of their own volition." "The headhunters came to you themselves through a website." = personally. This use of "themselves" emphasises "the headhunters". Specifically, they contacted you directly and did not use an agent or go-between.

"on their own" and "themselves" have the same meaning, they mean "alone, without any help" in this sentence you can use both. "do your homework on your own" "do your homework yourself" this means no one will help you, you have to do it alone.