I can't imagine that at 14 months she understands that it's the food making her feel unwell, so personally wouldn't think she is deliberately avoiding eating and drinking. It will take her good fortnight on the gluten free diet to get her to the stage of feeling well again (I know from experience) and a good longer than that to bump her up to optimal weight.
Given that she is so thin, you are going to have fuss over her diet like a fusspot for the next few weeks.
Personally I would start by getting some calories into her, in any way she'll take them, then concentrate on the vits and minerals when she is perking up. I sincerely hope that your paediatrician is advising you on feeding, and that she is having frequent check-ups, as she has probably been nutritionally deficient since weaning.
I would think that the optimum food for a child of that age with such a problem would be breast milk. I don't suppose there's a chance you're still lactating, is there? Or do you have a bf sister of friend who could donate a lot of milk?
Failing that, try her with fruit smoothies with banana, and honey or sugar, just to some calories into her. With some bananas, and some raspberries, bluberries, a splash of juice (one of those sticky mango or pineapple or peach nectars? or apple) and generous amount of honey. If she keeps this down that a good start, and you good start her on her gluten free food.
Is she able to take rice, or is that out too? My friend always swears by rice water for her children with upset stomachs, but it obviously depends on whether she can tolerate it.
Also CHECK the yoghurt ingredients- many have gluten in as a thickener!