Legal to use an existing business name?

If they haven't trademarked their name then you should check to see if it has been trademarked by anyone else. If it hasn't been then it is public property until it is trademarked. No need to contact them. In fact, if the name hasn't been trademarked by anyone it would be in your best interests not to altert them to that fact. That would also leave you time and opportunity to trademark it yourself.