Program.cs

class Program
{
    static void Main(string[] args)
    {
        var titles = GetTitles("Category:日本語の姓");
        File.WriteAllText("日本語の姓.txt", string.Join("\r\n", titles), Encoding.UTF8);
    }

    private static List<string> GetTitles(string cmtitle)
    {
        var titles = new List<string>();
        var client = new RestClient("https://ja.wikipedia.org/w/") {ThrowOnAnyError = true};
        string cmcontinue = null;
        do
        {
            var request = new RestRequest("api.php", DataFormat.Json)
                .AddParameter("action", "query")
                .AddParameter("list", "categorymembers")
                .AddParameter("cmprop", "title") // タイトルのみ取得
                .AddParameter("format", "json")
                .AddParameter("cmlimit", "500")
                .AddParameter("cmtitle", cmtitle);

            if (cmcontinue != null)
                request.AddParameter("cmcontinue", cmcontinue);

            var response = client.Get(request);
            var o = JObject.Parse(response.Content);
            titles.AddRange(o["query"]?["categorymembers"].Select(x => x["title"].ToString()));
            cmcontinue = o["continue"]?["cmcontinue"]?.ToString();
        } while (cmcontinue != null);

        return titles;
    }
}