可以使用以下代码示例按顺序重启IIS的Web API：

using System;
using System.Diagnostics;

public class IisHelper
{
    private const string IisResetCommand = "iisreset";
    private const string IisResetArgs = "/stop";

    public void RestartWebApi()
    {
        StopWebApi();
        StartWebApi();
    }

    private void StopWebApi()
    {
        var processInfo = new ProcessStartInfo
        {
            FileName = IisResetCommand,
            Arguments = IisResetArgs,
            CreateNoWindow = true,
            RedirectStandardOutput = true,
            UseShellExecute = false
        };

        var process = new Process { StartInfo = processInfo };

        process.Start();
        process.WaitForExit();

        Console.WriteLine("Web API stopped.");
    }

    private void StartWebApi()
    {
        var processInfo = new ProcessStartInfo
        {
            FileName = IisResetCommand,
            CreateNoWindow = true,
            RedirectStandardOutput = true,
            UseShellExecute = false
        };

        var process = new Process { StartInfo = processInfo };

        process.Start();
        process.WaitForExit();

        Console.WriteLine("Web API started.");
    }
}

可以使用以下代码调用 RestartWebApi 方法来按顺序重启Web API：

var iisHelper = new IisHelper();
iisHelper.RestartWebApi();

这将首先停止Web API，然后启动Web API。请确保代码具有适当的权限来执行IIS重启操作。