Both are correct and we use them but It's more common to say "¿y quién va a hacer la comida/el desayuno/la cena?" Because if you say "¿y quién va a cocinar?" people will understand that this person will be the one who will cook all the meals. For example if you go on holiday with your friends and you ask "¿y quién va a cocinar?" your friends will think that this person will cook all the meals of your holiday. You could say "¿y quién va a cocinar hoy?" and then they will think that this person will cook all the meals of today. This is the reason why we usually say "¿y quién va a hacer la comida/cena?". We say "¿y quién va a cocinar ahora?" too, but it's more common the other way. I hope it helps. Excuse my English, please. I have to learn a lot.

Muchas gracias, Camilo!

Otra opción es usar el verbo preparar. ¿Y quién va a preparar la comida/ la cena?

Hello. When you use the verb "hacer", you have to specify the subject, you have to say what thing should be done... "Y quien va a hacer la comida? ". If you prefer to use the verb "cocinar", the subject can be omited, because you always cooks a meal... "Y quien va a cocinar?". I prefer "Y quien va a cocinar?"