Is there a built in option in magento of emailing the client after submitting the contact form?
I can't find anything like it..
No, there isn't (as of Magento CE 1.9.1). If you look at the code that sends the contact form email in
Mage_Contacts_IndexController::postAction() you will see that there is not much room for customization either (except if you want to write your own controller action)
But writing an own module for this is not hard for an experienced Magento developer. It's either writing your own controller action as mentioned before, or writing an observer for the
contacts_index_post_postDispatch event that checks if a success message has been triggered and then sends another mail with its own email template to the email address in
Warning: Depending on how you write this, it could be misused for spam because at this point the email address is not verified and can be anything. Make sure not to include any content of the contact form in the email, keep it generic!