Well, there are a few different ways of doing that. One such way would be to hard code the form in HTML and have a form handler in PHP for example, to email the form data to an email address or insert it in to a MySQL database, or flat file such as a text document. If you are fluent in HTML, JavaScript, PHP and MySQL , it should be fairly simple. Even if you have a basic understanding, there are some good tutorials out there that can guide you through the process.

However, there are tools available to generate forms without any coding knowledge. There is one in Softaculous called FormGenerator that may be useful to you as well.

The bottom line is, if they want it to be self hosted rather than a remote form service, the form needs to be created from something and loaded on to their server.

Without knowing the complexity of the forms required, I can only speculate as to what may be helpful.

Email Form - Simple Form to Email PHP Contact Form
https://www.google.com/#q=php+mysql+form

Hopefully that will get you started.

Feel free to come back with any follow up questions you might have, we would be happy to assist you.