How do I implement SSL in Delphi to connect to a REDCap API server?
You are setting up the TLS connection correctly (provided the appropriate OpenSSL DLLs are available where Indy can find them).
What you are not setting up correctly is your data parameters. Curl's --data-urlencode command puts the data in the HTTP request body, not in the HTTP headers. So you need to put the data in the TStringList that you are posting (TIdHTTP will handle the url-encoding for you).
Try this instead:
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;


I am trying to use XE7 to connect to an in-house REDCap server. REDCap has a detailed description of the API at https://education.arcus.chop.edu/redcap-api/ and a test server at https://bbmc.ouhsc.edu/redcap/api  with a test token key.  There is assistance at https://mran.microsoft.com/snapshot/2015-08-18/web/packages/REDCapR/vignettes/TroubleshootingApiCalls.html in R.
I can connect to the test site with Curl and PostMan. My problem is how to implement this in Delphi with SSL.
The Curl script from PostMan:
curl --location 'https://bbmc.ouhsc.edu/redcap/api/' \
--data-urlencode 'token=9A81268476645C4E5F03428B8AC3AA7B' \
--data-urlencode 'content=record' \
--data-urlencode 'action=export' \
--data-urlencode 'format=csv' \
--data-urlencode 'rawOrLabel=label'

After much searching, this is my Delphi code. What have I missed? IdLogFile1 is a component on the form.
function TForm1.IdSSLIOHandlerSocketOpenSSL1VerifyPeer(Certificate: TIdX509; AOk: Boolean; ADepth, AError: Integer): Boolean;
begin
   showmessage('at  IOhandler');                      
     Result := true;                             // always returns true
end;


procedure TForm1.idHTTP2BtnClick(Sender: TObject);
var post      : string;
    Params    : TStringList;
    idHTTP    : TIdHTTP;
    SSL1      : TIdSSLIOHandlerSocketOpenSSL;
    status    : integer;
    response : TstringStream;
begin
   params   := TStringList.Create;
   idHTTP   := TIdHTTP.Create(nil);
   SSL1     := TIdSSLIOHandlerSocketOpenSSL.Create(idHTTP);
   response  := TstringStream.create;


   SSL1.SSLOptions.Mode        := sslmClient ;
   SSL1.SSLOptions.SSLVersions := [sslvTLSv1, sslvTLSv1_1, sslvTLSv1_2 ];// [  sslvSSLv3,  sslvSSLv23,sslvSSLv2, sslvTLSv1, sslvTLSv1_1, sslvTLSv1_2];
   SSL1.SSLOptions.VerifyDepth := 0;
   SSL1.OnVerifyPeer           := IdSSLIOHandlerSocketOpenSSL1VerifyPeer;
   SSL1.SSLOptions.VerifyMode  := [ ];
   idHTTP.IOHandler            := SSL1;

   memo1.Lines.clear;

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

   try

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

     IdHttp.Request.CustomHeaders.Clear;

 
     IdHttp.Request.CustomHeaders.Values['token']          := '9A81268476645C4E5F03428B8AC3AA7B';
     IdHttp.Request.CustomHeaders.Values['content']        := 'record';
     IdHttp.Request.CustomHeaders.Values['action']         := 'export';
     IdHttp.Request.CustomHeaders.Values['format']         := 'csv';
     IdHttp.Request.CustomHeaders.Values['rawOrLabel']     := 'label';
     IdHttp.Request.CustomHeaders.Values['verify_ssl']     := 'false';
     IdHttp.Request.CustomHeaders.Values['ssl_verify']     := 'false'; //various verify options ?
     IdHttp.Request.CustomHeaders.Values['ssl_verifypeer'] := 'false';

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

     idHTTP.Post('https://bbmc.ouhsc.edu/redcap/api/', params, response );


   finally
        memo1.Lines.add(' ');
        memo1.lines.add(idHTTP.ResponseText);
        memo1.Lines.add(' ');
        status           := idHTTP.ResponseCode;
        memo1.Lines.Add('code: ' + inttostr(status));
 
        idhttp.Disconnect;
 
   end;
   Params.Free;
   SSL1.Free;
   idHTTP.Free;
   response.Free;
end;

