Twitter has launched a new embeddable button which you can place on your website and site visitors can send you direct message on twitter from your website.
This lets site owners, businesses add another customer support channel.
Here is how it works, select any icon on their website
Add your Twitter Handle or user ID
This is it, get the embed code and place on your website so site users can contact you on Twitter form your website.
The website address is https://publish.twitter.com