首先，在ASP.NET Core 7.0 Web API项目中添加对System.Web.dll的引用。在Visual Studio中，右键单击项目，选择“添加引用”>“浏览”，然后浏览到.NET Framework的安装目录，选择“System.Web.dll”并单击“添加”。

然后，还需要在Web API项目中添加一个名为“MembershipProvider”的类，该类将实现System.Web.Security.MembershipProvider类。该类中的所有方法（例如ValidateUser和CreateUser）都应使用System.Web.Security.Membership类执行相应的操作。

下面是一个示例MembershipProvider类：

using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading.Tasks;
using System.Web.Security;

namespace WebApi.Membership
{
    public class MembershipProvider : System.Web.Security.MembershipProvider
    {
        // Implement all required MembershipProvider methods here
        public override MembershipUser CreateUser(string username, string password, string email, string passwordQuestion, string passwordAnswer, bool isApproved, object providerUserKey, out MembershipCreateStatus status)
        {
            // Implement the CreateUser method here
        }

        public override bool ValidateUser(string username, string password)
        {
            // Implement the ValidateUser method here
        }

        public override string ApplicationName { get; set; }
        // Implement all other required properties here
    }
}

一旦创建了MembershipProvider类，就需要在ASP.NET Core 7.0 Web API的Startup.cs文件中配置MembershipProvider并将其添加到Web API应用程序中。可以使用以下代码：

using WebApi.Membership;

public void ConfigureServices(IServiceCollection services)
{
    // Register MembershipProvider with the application services
    services.AddSingleton();

    // Add authentication middleware
    services.AddAuthentication(options =>
    {
        options.DefaultScheme = "Bearer";
    })
    .AddJwtBearer(options =>
    {
        options.TokenValidationParameters = new TokenValidationParameters
        {
            ValidateAudience = false,
            ValidateIssuer = false,
            ValidateIssuerSigningKey = true,
            IssuerSigningKey = new SymmetricSecurityKey(Encoding.UTF8.GetBytes(Configuration["Jwt:Key"]))
        };
    });

    services.AddMvc();
}

注意，对于示例代码，还需要在appsettings.json文件中添加JWT密钥（在Web API应用程序中使用JWT）：

{
  "Jwt": {
    "Key": "my_secret_key_for_jwt"
  }
}

最后，就可以在API控制器中使用MembershipProvider类中的方法了。例如，以下是使用MembershipProvider.ValidateUser方法验证用户凭据的示例代码：

using System.Web.Security;
using Microsoft.AspNetCore.Authorization;
using Microsoft.AspNetCore.Mvc;

[Route("api/[controller]")]
[ApiController]
public class AccountController : ControllerBase
{
    private readonly MembershipProvider _membershipProvider;

    public AccountController(MembershipProvider membershipProvider)
    {
        _membershipProvider = membershipProvider;
    }

    [HttpPost]
    [AllowAnonymous]
    [Route("login")]
    public IActionResult Login(string username, string password)
    {
        bool isAuthenticated = _membershipProvider.ValidateUser(username, password);

        if (isAuthenticated)
        {
            // Generate and return JWT token
        }
        else
        {
            return Unauthorized();
        }
    }

    // Other account-related methods here
}

使用MembershipProvider类时，请务必记住将“using System.Web.Security”添加到每个文件的顶部。