I'm a website designer and am looking at maybe using Jotform for my website contact forms.

Hi,

We actually have many users that do the same thing.  Almost everything can be setup to the specifications of your clients.  Email notifications, design, and much more can all be setup to suit their needs.  You can then share the form with them:

http://www.jotform.com/help/232-How-to-Share-Forms-with-a-Sub-Account-User

Or keep control of the form itself and forward everything via email.

If you have any questions regarding specifics, let us know and we will be happy to help.