feat: add Vue 3 and Nuxt integration packages

[Jorgagu]

This PR adds Vue 3 and Nuxt integration packages for Ory Elements, bringing feature parity with the existing React/Next.js implementation.

New Packages

@ory/elements-vue - Vue 3 component library for Ory authentication UIs

• Complete set of UI components for Login, Registration, Recovery, Verification, Settings, and Consent flows
• Vue 3 Composition API with <script setup> syntax
• Full TypeScript support with proper type inference
• Default theme with Tailwind CSS v4 styling
• Internationalization support (80+ locales)
• Session management via SessionProvider
• Customizable components through slot overrides

@ory/nuxt - Nuxt 3/4 integration module

• Auto-imported composables: useOryLogin, useOryRegistration, useOryRecovery, useOryVerification, useOrySettings, useOrySession, useOryLogout, useOryFlowError
• Server-side proxy for secure cookie handling
• Auth middleware for route protection
• Environment-based SDK URL configuration (NUXT_PUBLIC_ORY_SDK_URL)
• SSR-compatible session hydration

Example Application

Complete Nuxt example app demonstrating all authentication flows at examples/nuxt-app.

Related Issue or Design Document

Fixes #581

Checklist

• I have read the contributing guidelines and signed the CLA.
• I have referenced an issue containing the design document if my change introduces a new feature.
• I have read the security policy.
• I confirm that this pull request does not address a security vulnerability.
  If this pull request addresses a security vulnerability,
  I confirm that I got approval (please contact security@ory.sh) from the maintainers to push the changes.
• I have added tests that prove my fix is effective or that my feature works.
• I have added the necessary documentation within the code base (if appropriate).

Further comments

The Vue/Nuxt packages follow the same architecture patterns as the React/Next.js implementation:

• Same component hierarchy and naming conventions
• Same context/provider architecture (using Vue's provide/inject)
• Same flow handling logic and state management
• Same theming system structure with customizable components
• Same utility functions and SDK helpers

All authentication methods are supported including password, passwordless (code), social login (OIDC), passkeys, WebAuthn, TOTP, and recovery codes.

This fixes the issue #581

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Latest commit: 6e9245b

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 55.79%. Comparing base (f3fad4d) to head (6e9245b).
⚠️ Report is 274 commits behind head on main.

Additional details and impacted files

@@             Coverage Diff             @@
##             main     #586       +/-   ##
===========================================
+ Coverage   42.43%   55.79%   +13.36%     
===========================================
  Files         136      173       +37     
  Lines        2008     3194     +1186     
  Branches      288      472      +184     
===========================================
+ Hits          852     1782      +930     
- Misses       1149     1344      +195     
- Partials        7       68       +61     

Components	Coverage Δ
@ory/elements-react	54.63% <ø> (+17.84%)	⬆️
@ory/nextjs	62.52% <ø> (-3.46%)	⬇️

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[Jorgagu]

Hi @jonas-jonas @aeneasr @vinckr, this PR is ready for review 🎉

👋 @Jannik-Hm @Saganic, could you please test this as well and let me know if I missed anything during my own testing?