K.A.
At 15 months old, the safest place for your daughter is in the cart with the seat belt on. She is probably too little to understand why you need to go the next aisle, etc. Running free she can pull stuff off of the shelves or get run over by someone else's cart. For now, you may have to go shopping after your significant other gets home, if that is an option. Take a sippy cup and a snack with you or try putting her in her stroller and just buying a smaller quantity of stuff that you can put in the stroller or a hand basket. At home, get her a small shopping cart and have her practice shopping so she knows what to expect at the store. When my daugter was a little older (2 or so) we would take her shopping cart to the store with us. It made the trip take longer, but she enjoyed putting items in her basket and helping.