Heads can ask, but my understanding is that teachers (possibly just union members) have the right to refuse. Support staff can refuse to cover if it is not part of their normal duties, again a good idea for them to be unionised though.
I am not convinced that agency staff will work on strike days in any large numbers- I know some will but many supply staff are NEU, and they can simply say they are not available that day. Of course, non NEU supply cannot be compelled to work on that day either.
Banks of supply staff were tried last winter too- I don't think they exist? Does anyone's school actually have this?
I would not want to be that volunteer, supervising a large group of students alone. What about first aid etc? What protection do the volunteers have if a child gets hurt or makes an accusation because they are not being properly supervised?
Heads also need to consider things like do they have enough first aiders on site, and can they evacuate the building in the event of a fire etc- they couldn't open just with volunteers. And I hope most heads will not take the attitude of opening at all costs- don't forget some heads are NEU, some NAHT voted to strike etc.
They also need to consider things like illness among support staff- if they are relying on, say, 5 staff to come in to open a school, what if one of them is ill? Not exactly unlikely this winter?
And heads should be reminded that they need to preserve their relationship with staff too- if they work really hard to essentially break the strike, they'll likely lose teaching staff. I wouldn't want to work for a head like that, I'd start looking for a new job ASAP.