You can use the ShellAPI funciton ShFormatDrive(). Example:

const SHFMT_DRV_A = 0;
const SHFMT_DRV_B =  ;
const SHFMT_ID_DEFAULT = $FFFF;
const SHFMT_OPT_QUICKFORMAT = 0;
const SHFMT_OPT_FULLFORMAT =  ;
const SHFMT_OPT_SYSONLY = 2;
const SHFMT_ERROR = - ;
const SHFMT_CANCEL = -2;
const SHFMT_NOFORMAT = -3;
function SHFormatDrive(hWnd : HWND;
                       Drive : Word;
                       fmtID : Word;
                       Options : Word) : Longint
   stdcall; external 'Shell32.dll' name 'SHFormatDrive';
procedure TForm .Button Click(Sender: TObject);
var
  FmtRes : longint;
begin
  try
    FmtRes:= ShFormatDrive(Handle,
                           SHFMT_DRV_A,
                           SHFMT_ID_DEFAULT,
                           SHFMT_OPT_QUICKFORMAT);
    case FmtRes  of
     SHFMT_ERROR : ShowMessage('Error formatting the drive');
     SHFMT_CANCEL : 
       ShowMessage('User canceled formatting the drive');
     SHFMT_NOFORMAT : ShowMessage('No Format')
    else
     ShowMessage('Disk has been formatted');
    end;
  except
  end;
end;