Not quite sure exactly what you mean by this Karell and I don't think there is an easy way to do this, but:
You can get the Verbot to startup with:
Verbot5Engine.exe --input="input text"
(where the --input command provides an initial 'spoken/typed' input.
Now, if you wanted the bot to make a spoken reply to something happening on your website, then provided that the "something" was translated into an input (to replace "input text") then you should start the Verbot and get a response.
Second, on startup the bot can execute a _startup rule. This allows you to decide on a sequence of "startup" events etc - depending on how you write the _startup rule.
Now, I know nothing about Website design; but it is also worth bearing in mind that the Verbot can accept input from a Speech-to-text program.
Therefore, in principle, it can accept typed input from an external source.
This suggests to me that you might be able to "fool" the bot into thinking that the input from your Website was STT input.
My final thought is that using c# vcm code, you could (in principle again) persuade the bot to accept input into a vcm function and use that to control the 'bots actions, since the 'bot normally expects to get data out of a vcm function.
I'm sorry to be a bit vague in my suggestions, but Wedsite-bot interfacing is definitely not something I've ever tried!