To add Facebook Chat to Google Sites, you can use this free tool. Follow these steps.
#1. First, access this free tool and click Create widget for free
#2. Choose Facebook Chat layout and click Continue with this template .
#3. Next, After connecting to your Facebook Account, select a Facebook page.
You can also set layout, style here then click Add to website for free.
#4. Next, Click Publish on top right
#5. Choose Free Plan and click Copy Code .
#6. You can add it to Google Sites via Embed Feature.
#7. Paste your embed code
#8. Result