Why does a 4 year old need a double bed? A single is cheaper, cheaper to provide linens for, cheaper to launder the linens for, easier for the child to manage (getting in and out and making their own bed) and leaves more room for play.
If VOCs are a concern you can buy the mattress, take off the wrappings and let it sit for a while before having your daughter sleep on it.
Proper ventilation of the home is necessary at all times to allow the chemicals in cleaners etc to disperse. If you live in an unpolluted environment (away from industry and major roads) then opening a bedroom window every morning for 5 minutes is a good idea.