[Doc] Add AI Badgr framework integration documentation

[miguelmanlyx]

Purpose

Add documentation for AI Badgr, an OpenAI-compatible LLM provider, in the framework integrations section. AI Badgr uses tier-based model naming (basic/normal/premium) and can work with vLLM as a backend or be accessed as a hosted service.

Test Plan

# Verify markdown linting
pre-commit run --hook-stage manual markdownlint --files docs/deployment/frameworks/aibadgr.md

# Verify documentation builds
API_AUTONAV_EXCLUDE=vllm mkdocs build

Test Result

• Markdown linting: ✅ Passed
• Documentation build: ✅ Generated site/deployment/frameworks/aibadgr/index.html
• Pattern consistency: ✅ Matches existing framework docs (LobeChat, Chatbox, etc.)

Changes

New file: docs/deployment/frameworks/aibadgr.md

Documents two integration scenarios:

1. vLLM as backend: Configure AI Badgr to use vLLM's OpenAI-compatible endpoint
2. AI Badgr as provider: Connect to AI Badgr's hosted service using tier models

Example usage with environment variables:

import os
from openai import OpenAI

client = OpenAI(
    base_url=os.getenv("AIBADGR_BASE_URL", "https://aibadgr.com/api/v1"),
    api_key=os.getenv("AIBADGR_API_KEY")
)

response = client.chat.completions.create(
    model="premium",  # tier models: basic, normal, premium
    messages=[{"role": "user", "content": "Hello"}]
)

Includes tier model mappings: phi-3-mini (basic), mistral-7b (normal), llama3-8b-instruct (premium).

---

Essential Elements of an Effective PR Description Checklist

• The purpose of the PR
• The test plan
• The test results
• Documentation update (new framework integration doc)
• Release notes update (not user-facing, docs-only change)

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[mergify]

Documentation preview: https://vllm--30669.org.readthedocs.build/en/30669/

[gemini-code-assist]

Code Review

This pull request introduces documentation for integrating AI Badgr as a framework. The new documentation is clear and follows the existing format for framework integrations. However, I've identified a significant correctness issue: the example commands use a model name that does not exist, which will cause them to fail. I've provided suggestions to replace it with a valid model name.