Most offer additional plug in cards for very basic on/off/fault/some remote setpoint adjustment capabilities which you can interface with your own controller relays.
Very few brands have built in web capabilities, though some can offer some computer software as a paid extra.
Those warehouse heat pumps were non inverter types from memory. I really doubt you would find any instantly compatible products for a non major brand product. What exactly are you aiming to do and can it be done with the remote/timers etc instead?