You can follow these steps to add Instagram Profile to Google Sites.
#1. First, access this free tool (Free version is enough) and click Create widget for free
#2. Choose one of Profile Widget layouts and click Continue with this template .
#3. Next, After connecting to your Instagram Account, select a Instagram 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