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Yes. The web chat is a channel like the others, so the widget on your site drops conversations into the same inbox and the same automations can answer them.
What that changes beyond convenience:
- One contact record. Someone who messaged on Instagram in March and lands on your site in June is the same person, with the same tags, rather than two strangers.
- The same flows. The keyword routing and the qualifying questions you already built work on the widget without being rebuilt.
- No messaging window. A web chat has no 24 hour rule, because it is your own site rather than Meta's platform.
Setup is a script tag on your site. The widget carries your colours and your greeting, and you choose whether it asks for an email before or after the conversation. Asking after gets you more conversations and fewer addresses, which is usually the better trade for a site with modest traffic.
Two things to decide. What the widget says when nobody is available, because a chat that promises a human and delivers silence is worse than no widget. And whether it should appear on every page, since a widget on the checkout page interrupts a purchase more often than it saves one.
On a free workspace the widget carries an attribution line, the same way free plan DMs do.
The Inbox basics article covers how conversations from different channels sit side by side and how assignment works across them.