All plants require free oxygen, and any (part of a) plant that is not actively photosynthesising has to obtain its oxygen from its environment. We owe our oxygen supply to the fact that, on average, plants produce more oxygen than they consume.
Plants respire in much the same way as other eukaryotes, relying on oxygen for the final step of their electron transport chain. Like us, they can switch to anaerobic respiration for a short time to cope with low-oxygen conditions, but cannot do this indefinitely.
For example, land plants in a flood suffer a shortage of oxygen to their roots. The plants might survive for hours or days, but will eventually die unless the water recedes.
Similarly, aquatic plants will die if the water they grow in has too little oxygen. This can happen even when the surface of the water is in bright sunlight, if deeper plants are shaded out by a bloom of algae. The total oxygen demand exceeds the supply, and most or all of the plants die.
A cabbage kept in the dark in an oxygen-free environment will asphyxiate.