I do agree with the responses you've gotten so far, she'll probably be fine. But I was never one to put much stock in the odds, especially with one SIDS death in our family history. I didn't sleep easy (literally) until each baby could not only turn over, but showed me she had the ability to do a good strong 'push-up.' Then I felt pretty safe.
I did use a positioner and had no problems with my babies getting stifled by it. Binding it tightly under her sheet prevented her moving it out of position.
A maternity nurse taught me a good trick with my second daughter -- use a cloth diaper or the sheets of the crib to tightly tuck baby into place. The diaper or sheet was folded into a long narrow rectangle, placed across baby's tummy/chest and then tucked tightly under each side of the mattress. My babies could wiggle and kick and suck their fists, but they couldn't turn over. The cloth extended far enough beneath each side of the mattress that the weight of baby + mattress prevented the cloth being pulled out. The key was to make sure all loose edges were folded within the sheet (so baby's legs and arms only touch tight, clean folds, not loose edges that can be pulled at). I also woke up every couple of hours to check -- or feed -- but I rarely had much fixing to do.
Even better, I believe they make a positioner that uses exactly the same principle but actually goes right round the mattress underneath. There are also the SIDS monitors, etc, but they can get pretty pricey.
But even with family history lurking, my babies made it through infancy just fine! :-)