Şöyle olabilir

private void verileriGonder() {
  string userAgent = "VBA6.0.SPECIAL-APP";
  Dictionary<string, string> veriler = new Dictionary<string, string>();
  veriler.Add("proccessor", Text12.Text);
  veriler.Add("hdd", Text9.Text);
  veriler.Add("mainboard", Text10.Text);
  veriler.Add("ram", Text11.Text);
  veriler.Add("serialnumber", Text5.Text);
  veriler.Add("user", Text3.Text);
  veriler.Add("graphiccard", Text8.Text);
  veriler.Add("label", Text4.Text);
  veriler.Add("ipaddress", Text1.Text);
  veriler.Add("ipgateway", Text2.Text);
  veriler.Add("dns1", Text7.Text);
  veriler.Add("dns2", Text6.Text);
  string gelenMesaj = MakeWebRequest("POST", "http://adres.comapi.php", "computerinfo=" + jsonGenerator(veriler), userAgent);
  if (gelenMesaj == "unsuccess") {
    MessageBox.Show("Veritabanına ekleme/güncelleme işlemi sırasında hata oluştu!");
  }
}

public string jsonGenerator(Dictionary<string, string> veriler) {
  string jsonString = "{" + Environment.NewLine;
  int counter = 1;
  foreach(string indexKey in veriler.Keys) {
    jsonString = jsonString + "\"" + indexKey.Trim() + "\":\"" + veriler[indexKey].Trim() + "\"";
    if (counter != veriler.Count) {
      jsonString = jsonString + "," + Environment.NewLine;
    }
    counter++;
  }
  jsonString = jsonString + Environment.NewLine + "}";
  return jsonString;
}

public string MakeWebRequest(string method, string url, string post_data, string user_agent) {
  HttpWebRequest req = (HttpWebRequest) WebRequest.Create(url);
  req.Method = method;
  req.UserAgent = user_agent;
  req.Timeout = 60000;
  req.ReadWriteTimeout = 60000;
  req.KeepAlive = true;
  req.AllowAutoRedirect = true;
  if (method == "POST") {
    req.ContentType = "application/x-www-form-urlencoded";
    using(var streamWriter = new StreamWriter(req.GetRequestStream())) {
      streamWriter.Write(post_data);
      streamWriter.Flush();
      streamWriter.Close();
    }
  }
  string responseString = "";
  using(HttpWebResponse response = (HttpWebResponse) req.GetResponse()) {
    using(var streamReader = new StreamReader(response.GetResponseStream())) {
      responseString = streamReader.ReadToEnd();
    }
  }
  return responseString;
}