Usually the Ballad of Gales is required to enter Mother and Child Isles and obtain the Fire and Ice Arrows. However there multiple methods of getting into Mother and Child Isles without the Ballad of Gales. Currently this only has use in a Low% speedrun as getting the Ballad of Gales is faster for Any%.
With enough speed, you can simply superswim into Mother and Child Isles while the island isn't loaded and then kill your superswim speed to stay inside the island while it loads.
Alternatively, you can also Zombie Hover from the outside portion of the island into where the fairy queen is and then die on the inside. Link will respawn inside the island due to his death location.
The easiest way of exiting Mother and Child Isles without the Ballad of Gales is to Item Slide with a somewhat precise angle and clip through the floor.
As with getting in, you can also Zombie Hover out of Mother and Child Isles and die on the outside portion of the island to respawn there.