A standard intervalometer can work for HDR time-lapse, but it depends on how your camera handles bracketing.

For basic HDR time-lapse, the usual approach is to set auto-bracketing in the camera and have the intervalometer trigger each bracket set at your chosen interval. If the camera supports shooting the whole bracket sequence in a continuous burst, you can get the desired pattern of multiple bracketed exposures followed by a wait.

If the camera requires separate shutter releases for each bracketed frame, there may be pauses between exposures instead.

The main reason some intervalometers advertise HDR support is that they can better control bracket sequences, especially in bulb mode or with longer exposures. A more advanced controller may be able to program separate HDR exposures directly, while a basic intervalometer mostly just triggers the shutter at intervals.

So: for short exposures, a normal intervalometer plus in-camera bracketing/continuous mode may be enough. For long exposures or bulb-mode HDR sequences, an HDR-capable controller can be more flexible and reliable.