Paste the tag before </body> in the layout all your pages use, usually src/layouts/Layout.astro:
<script is:inline async src="https://sendshot.dev/widget.js" data-key="YOUR_PROJECT_KEY"></script>
Replace YOUR_PROJECT_KEY with the key from Project → Settings → Installation in your Sendshot dashboard.
is:inline tells Astro to leave the tag as it is instead of bundling it.
Check it works
Open your site. The feedback button appears in the bottom-right corner after the page has loaded. Press Cmd/Ctrl + Shift + F to open it from the keyboard, send a test report, and it shows up in your dashboard.
No button? Make sure the domain you're on is listed in the project's allowed domains.