До слова implementation напишите такой код:

{tapi Errors}
const TAPIERR_CONNECTED = 0;
const TAPIERR_DROPPED = -1;
const TAPIERR_NOREQUESTRECIPIENT = -2;
const TAPIERR_REQUESTQUEUEFULL = -3;
const TAPIERR_INVALDESTADDRESS = -4;
const TAPIERR_INVALWINDOWHANDLE = -5;
const TAPIERR_INVALDEVICECLASS = -6;
const TAPIERR_INVALDEVICEID = -7;
const TAPIERR_DEVICECLASSUNAVAIL = -8;
const TAPIERR_DEVICEIDUNAVAIL = -9;
const TAPIERR_DEVICEINUSE = -10;
const TAPIERR_DESTBUSY = -11;
const TAPIERR_DESTNOANSWER = -12;
const TAPIERR_DESTUNAVAIL = -13;
const TAPIERR_UNKNOWNWINHANDLE = -14;
const TAPIERR_UNKNOWNREQUESTID = -15;
const TAPIERR_REQUESTFAILED = -16;
const TAPIERR_REQUESTCANCELLED = -17;
const TAPIERR_INVALPOINTER = -18;

{tapi size constants}
const TAPIMAXDESTADDRESSSIZE = 80;
const TAPIMAXAPPNAMESIZE = 40;
const TAPIMAXCALLEDPARTYSIZE = 40;
const TAPIMAXCOMMENTSIZE = 80;
const TAPIMAXDEVICECLASSSIZE = 40;
const TAPIMAXDEVICEIDSIZE = 40;

function tapiRequestMakeCallA(DestAddress : PAnsiChar;
AppName : PAnsiChar;
CalledParty : PAnsiChar;
Comment : PAnsiChar) : LongInt;
stdcall; external "TAPI32.DLL";

function tapiRequestMakeCallW(DestAddress : PWideChar;
AppName : PWideChar;
CalledParty : PWideChar;
Comment : PWideChar) : LongInt;
stdcall; external "TAPI32.DLL";

function tapiRequestMakeCall(DestAddress : PChar;
AppName : PChar;
CalledParty : PChar;
Comment : PChar) : LongInt;
stdcall; external "TAPI32.DLL";

Нажатие кнопки обработайте следующим образом:

procedure TForm1.Button1Click(Sender: TObject);
var
  DestAddress : string;
  CalledParty : string;
  Comment : string;
begin
  DestAddress := "000-00-00"; {phone number}
  CalledParty := "goldfaq";
  Comment := "Calling to goldfaq";
  tapiRequestMakeCall(pChar(DestAddress),
  PChar(Application.Title),
  pChar(CalledParty),
  PChar(Comment));
end;