You can actually add a chatbot into your website very easily using Dialogflow.
When you go to the Integrations tab, you will see the following option:
To add the integrated web demo, all you really have to do is to copy the HTML provided in the web demo integration popup window and add it into your website.
For example, here is a bot I built using the web demo integration:
However, the 1-click web demo integration has quite a few limitations.
There are other limitations, but these are the ones people frequently want to do.