Migrate Shopify Inbox to the official <shopify-chat> web component

[paul-phan]

Context

Hydrogen's July 30, 2026 developer preview adds official Shopify Inbox support through ShopifyScripts and the <shopify-chat> web component:

<ShopifyScripts
  shop={{shopId, storefrontId, myshopifyDomain}}
  inbox
/>
<shopify-chat />

Release notes: https://hydrogen.shopify.dev/update/developer-preview-release-notes-july-30-2026

Pilot currently uses the legacy shopifyChatV1.js loader in app/components/shopify-inbox.tsx, requires PUBLIC_SHOPIFY_INBOX_SHOP_ID, and relies on private widget selectors for its custom trigger. This was introduced in #338 and the overlay behavior was added in #447.

Do not implement this against @shopify/hydrogen@preview. The preview is incompatible with the current Pilot stack. Start once Shopify publishes the API in a stable Hydrogen release with migration guidance.

Scope

• Replace the legacy loader with ShopifyScripts using inbox.
• Render <shopify-chat> in the root layout.
• Remove PUBLIC_SHOPIFY_INBOX_SHOP_ID, legacy URL/query construction, and obsolete setup documentation.
• Preserve the overlay visibility behavior and the custom "Message us" action using Shopify's documented public web-component API only.
• Verify CSP, nonce handling, consent, analytics, locale, and client-side navigation.
• Update .weaverse/docs/shopify-inbox.md, .env.example, and relevant types.

Acceptance criteria

• Uses a stable @shopify/hydrogen release that officially exports ShopifyScripts.
• Inbox loads through <ShopifyScripts inbox> and <shopify-chat> without the legacy loader or Inbox-specific shop ID.
• The widget remains hidden while Pilot drawers, dialogs, and popups are open.
• The custom "Message us" action opens chat without private Shadow DOM selectors.
• Missing or disabled Inbox configuration degrades safely without breaking the storefront.
• Verified on a deployed domain on desktop and mobile with no CSP or console errors.
• Verified AI-agent chat and staff handoff with "Require sign-in to chat with staff" disabled.
• Build, typecheck, and relevant tests pass.
• Setup and migration documentation is updated.