No. HttpWebRequests are not reusable.
Just move the creation of your WebRequest into the body of your loop:
string result ="";
bool callComplete = false;
while (callComplete != true)
{
var request = (HttpWebRequest)WebRequest.Create(target);
request.ContentType = "application/json";
request.Headers.Add("Authorization", "Bearer " + oAuthKey);
request.AllowAutoRedirect = false;
//...