@Darcy78 As this is an international forum, with hosts from all over the world, you might have more luck by seeing if there is an Airbnb Facebook group for Melbourne.
And I assume you are just trying to research this for the mom and her baby- you can't book an Airbnb for someone else- third party bookings are against the Airbnb TOS. If you find something, unless you are also going to be a guest on this booking, she'll have to book under her own account.
But even if you find a host who says they have a suitable listing, make sure to be really specific about the extent of the allergies, what things would trigger them, etc. One of the problems for Airbnb hosts in trying to accommodate something like this, unless this is exactly what the hosts cater to as a business model, is that it would be stressful for the host to try to make sure there wasn't a spot of dust somewhere, or that they hadn't used a cleaning product, or anything else that would cause the child to have a reaction. Then the host is faced with an ill child, an upset mother, a cancellation and demand for a refund, and possibly a bad review. It's a lot to ask of a host, unless they have the same sort of allergy issues and therefore would know exactly how to make sure all would be non-threatening to the guest.
