Dropbox Business API 'bad request' error

BTW, this code is untested, but I think you can simplify to this:

var webRequest = (HttpWebRequest)WebRequest.Create("https://api.dropbox.com/1/team/groups/list");
webRequest.Method = "POST";
webRequest.ContentType = "application/json";
webRequest.Headers.Add("Authorization: Bearer <accesToken>"); 

using (var streamWriter = new StreamWriter(webRequest.GetRequestStream()))
{
    streamWriter.Write(JsonConvert.SerializeObject(bodyContent));
}

var webResponse = (HttpWebResponse)webRequest.GetResponse();

FYI, in the case of /groups/list, the JSON body should just be {}, since this method doesn't take any parameters.