You could use the same code that you have in your existing website. However, you'll see that the form actio points to a PERL (.pl) script in the CGI-bin folder on the server. You need such a mail script in the CGI-bin of the new server (and amend te code to point at the correct scrit). A popular one is FormMail (use Gogle and find a free download).
However, there is a danger (apparently) in using these scripts as they can get hijacked by spammers to send vast amounts of spam and then your ISP may suspend your account. If you know, say, PHP you can write a page that will send such a contact form that doesn't use a separate script in CGI-bin. This would be a safer bet.