JotForm works great with Sites. Easy to build and embed into a page. Can be configured to send emails upon successful form submission. A free version is available if you have less than 100 submissions per month.

You can add payment forms using PayPal and many other services.

Get the full source code and insert it using the code section of the Embed function.

If you modify the form you must get the new source code and insert it again.

Instead of embedding it you can link to it by using the URL option in the Embed edit function.

Another method is to create a Web App. To create Web App for this see these instructions and see my Web App code.