What type of site do you have? PHP, Perl, ASP, or something else? If you want you can just put a link to a registrar. Or you can perform a whois check and then register the domain. It all depends on how deep you want to go.
Well, with straight HTML it will not be possible. With ASP it would be possible to code a whois checker. You could then interface with a registrar and register domains for your customers. It will take lots of work if you are coming from straight HTML.
Your best bet is to place a form on your site that posts to another site which currently has all the code done of registering done. They will give you credit and give you a kick back.
But if you really want to do all that work don't let that stop you from coding the whole thing yourself. It can be very fun. :) I have done it several times for several companies over the years.