If my newborn won’t sleep on her back, will she get sids?

Probably not. Fortunately, very few babies die of sids. Years ago in america, when babies slept on their tummies, about 3 babies out of every 1000 would die of sids. Nowadays, when babies sleep on their backs, about 1 to 1.5 babies per 1000 die of sids. So, the chances of not getting sids is better if sleeping on the back. Parents should talk with the doctor if a baby won't sleep on his/her back.
The risk is higher.. ...With sleeping on her stomach, and higher yet with co-sleeping. Sleeping on the side isn't as safe as on the back, but may be worth trying if "back-to-sleep" totally fails. Otoh, is it really "won't sleep", or just normal colic? Your doctor should be able to tell the difference.
Maybe! . We know that back sleeping is a significant risk reducer in sids deaths in babies. It is the single most important thing you can do to prevent death from sids. If your baby is less six months old, keep putting him down on his back to sleep. Also breastfeeding, not smoking, keeping the room a bit cool (think 68 not 74), not wearing a hat, are other factors found to reduce risk.