코드 자체는 문제 없어보입니다.
제가 추가로 살을 붙여서 해 본 코드로는 문제없이 결과가 받아지는데요.
한 번 확인해 보세요.
Dim url As String = "https://dapi.kakao.com/v2/local/search/address.json?query=" + "강남동"
Dim resPonse As HttpWebResponse = Nothing
Dim Reader As StreamReader = Nothing
Dim webResponse = ""
Dim Request As HttpWebRequest = DirectCast(HttpWebRequest.Create(url), HttpWebRequest)
Request.Headers.Add("Authorization", "KakaoAK " & "YOUR_API_KEY"
Request.AllowAutoRedirect = False
Request.ContentType = "application/json;charset=UTF-8"
Request.Method = "GET"
resPonse = DirectCast(Request.GetResponse(), HttpWebResponse)
Dim webStream As Stream = resPonse.GetResponseStream()
Reader = New StreamReader(webStream)
While Reader.Peek >= 0
webResponse = Reader.ReadToEnd()
End While
Console.WriteLine(webResponse)