您正在正确设置TLS连接（前提是Indy可以找到 * 适当的 * OpenSSL DLL）。
您没有正确设置的是数据参数。Curl的--data-urlencode命令将数据放在HTTP请求正文中，而不是HTTP头中。因此，您需要将数据放入正在发布的TStringList中（TIdHTTP将为您处理URL编码）。
试试这个：

procedure TForm1.idHTTP2BtnClick(Sender: TObject);
var
  params    : TStringList;
  idHTTP    : TIdHTTP;
  idSSL     : TIdSSLIOHandlerSocketOpenSSL;
  status    : integer;
  response  : string;
begin
  params := TStringList.Create;
  try
    idHTTP := TIdHTTP.Create(nil);
    try
      idSSL := TIdSSLIOHandlerSocketOpenSSL.Create(idHTTP);    

      idSSL.SSLOptions.Mode        := sslmClient ;
      idSSL.SSLOptions.SSLVersions := [sslvTLSv1, sslvTLSv1_1, sslvTLSv1_2 ];
      idSSL.SSLOptions.VerifyDepth := 0;
      idSSL.OnVerifyPeer           := IdSSLIOHandlerSocketOpenSSL1VerifyPeer;
      idSSL.SSLOptions.VerifyMode  := [ ];
      idHTTP.IOHandler := idSSL;

      Memo1.Lines.Clear;

      idHTTP.ReadTimeout                 := 3000;
      idHTTP.ConnectTimeout              := 3000;
      idHTTP.Request.BasicAuthentication := false;

      try    
        idHTTP.HandleRedirects := true;
        idHTTP.Intercept       := IdLogFile1;
        IdLogFile1.Active      := true;

        params.Add('token=9A81268476645C4E5F03428B8AC3AA7B');
        params.Add('content=record');
        params.Add('action=export');
        params.Add('format=csv');
        params.Add('rawOrLabel=label');

        idHTTP.Request.ContentType := 'application/x-www-form-urlencoded';
        idHTTP.Request.Charset     := 'utf-8';
        idHTTP.HTTPOptions         := [hoKeepOrigProtocol, hoForceEncodeParams];

        response := idHTTP.Post('https://bbmc.ouhsc.edu/redcap/api/', params);    
      finally
        Memo1.Lines.Add(' ');
        Memo1.Lines.Add(idHTTP.ResponseText);
        Memo1.Lines.Add(' ');
        status := idHTTP.ResponseCode;
        Memo1.Lines.Add('code: ' + IntToStr(status));
      end;
    finally
      idHTTP.Free;
    end;
  finally
    params.Free;
  end;
end;

两个更简单的替代方案是System.Net.HttpClient和mORMot。两者都按预期工作，结果相同。用 Delphi 11测试。

program Project1;

{$APPTYPE CONSOLE}

uses
  System.SysUtils,
  System.Classes,
  System.Net.HttpClient,
  Syncommons,
  Syncrtsock;

var
  http : TWinHttp;
  ResHeader,ResData : SockString;
  Status : cardinal;

  Client: THTTPClient;
  Response: IHTTPResponse;
  params : TStringList;
begin
    /// mORMot alternative
    http := TWinHttp.Create('bbmc.ouhsc.edu','443',true);
    try

      Status := http.Request('/redcap/api/',
                             'POST'
                             ,0,
                             'Content-Type: application/x-www-form-urlencoded',
                             'token=9A81268476645C4E5F03428B8AC3AA7B&content=record&action=export&format=csv&rawOrLabel=label',
                             '',
                             ResHeader,
                             ResData);
    finally
      http.Free;
    end;
    writeln(formatUTF8('mORMot -> Status= % Response=%',[Status , resdata ]));

    /// HttpClient alternative
    Client := THTTPClient.Create();
    try
      params := TStringList.Create;
      try
        params.Add('token=9A81268476645C4E5F03428B8AC3AA7B');
        params.Add('content=record');
        params.Add('action=export');
        params.Add('format=csv');
        params.Add('rawOrLabel=label');
        Client.CustHeaders.Add('Content-Type' , 'application/x-www-form-urlencoded');
        Response := Client.Post('https://bbmc.ouhsc.edu/redcap/api/', params);
        writeln(formatUTF8('HttpClient -> Status= % Response=%',[Response.StatusCode, Response.ContentAsString]));

        Assert(Resdata = Response.ContentAsString); // checked!
      finally
        params.Free;
      end;
    finally
      Client.Free;
    end;
    readln;
end.