My first thought: Slight allergy to detergents/softeners. Use a totally allergen-free ("free and clear" or "hypoallergenic") detergent -- with no colors; no scents at all; etc. Switch right away and wash things in it a couple of times before you put them on the bed. If she uses a mattress cover, wash that in the new detergent as well. Same applies to whatever she's wearing to sleep -- rewash a few times in the dye-free, scent-free, allergen-free detergent.
Does she use a pillow? Drop that -- it might be the pillow or its filling or the pillowcase that is giving her a contact allergy. She's kind of young for a pillow anyway. Eliminate it and see if it helps. The fact that the rash is on the face and neck might point to the pillow as a culprit.
If you're thinking, "She has slept on the same sheets/pillow for a while and hasn't had any reactions" -- be aware that allergies can appear at different times, and someone who has had no reactions previously can suddenly start having reactions at any time.
You said the rash came after her first time eating raw apple -- good catch on your part. It could be a food allergy (and yes, one can be allergic to apples; my friend is horribly allergic to them). If she has had no further apples, I'm not sure about it, but ask the doctor. Has she had any other new foods recently?
Heat rash could be another issue but I would expect that on the body and not on the face.
And do ask the doctor's office. Does your pediatrician have a "nurses' advice line" where you can call and just ask a question rather than taking her in for an appointment? I'd start with a call but if they want to see her, do go, and ask to schedule it right after her nap time so they can see the rash before it fades.