It seems not possible with the "standard"
Telegram widgets to integrate a complete Telegram channel with live chat directly into the Website, but only such things like
embed single posts with a link to open the channel in Telegram etc..
But possibilities are not limited by that, as the
Telegram API is open to use for writing one's own application with all possible functions of Telegram adjusted to one's needs. For example,
here is the source code to the official Telegram web client which runs at web.telegram.org. One could rebuild the functionality of that or whatever one needs into one's own website. So to integrate an embeded Telegram channel/group with livechat as replacement for the shoutbox should be possible with that somehow, for example.
The
Login widget seems interesting and could be great for mobile phone users to associate an account with Telegram instead of e-mail to log-in here, but would require much change to the database code here. So not totally quick and easy to implement but surely somehow possible.
I will try some experiments, but maybe not yet something quickly visible today.