.NET: Use Responses API for hosted web search in AG-UI - #7843

Motivation & Context

HostedWebSearchTool uses web_search_options with Chat Completions. Azure OpenAI deployments can reject that parameter with HTTP 400. The AG-UI server sample needs the Responses API wire format, where web search appears in the tools array.

Description & Review Guide

• What are the major changes? The AG-UI server now uses the OpenAI SDK directly with BearerTokenPolicy and a Responses client. The sample no longer depends on Azure.AI.OpenAI. Documentation now explains the Azure OpenAI v1 endpoint and distinguishes direct inference from a Foundry project. Unit coverage captures the outbound request and confirms tools[].type is web_search while web_search_options is absent.
• What is the impact of these changes? Hosted web search uses the Azure OpenAI Responses API wire format and avoids the unsupported parameter reported in issue .NET: [Bug]: Request processing fails when web search is enabled because of an unsupported web_search_options parameter #7578.
• What do you want reviewers to focus on? Review the direct OpenAI SDK authentication, Responses client selection, and outbound request shape.

IChatClient chatClient = new OpenAIClient(
        new BearerTokenPolicy(new DefaultAzureCredential(), "https://ai.azure.com/.default"),
        new OpenAIClientOptions { Endpoint = new Uri(endpoint) })
    .GetResponsesClient()
    .AsIChatClient(deploymentName);

Related Issue

Fixes #7578

Contribution Checklist

• The code builds clean without any errors or warnings
• All unit tests pass, and I have added new tests where possible
• The PR follows the Contribution Guidelines
• This PR is linked to an issue and there is no other open PR for this issue (see Related Issue above).
• This is not a breaking change.