Small plastic container suited to the size of the critter, carried in a backpack and later placed in a larger food cooler chest is what most of my friends use on field hunts. Damp paper towels can be used when containing aquatic animals. Small holes can be punched or drilled into the containers since little air is required until you release them or get them home.
Pillow cases are the standard for most herpers who collect snakes and almost anything else.
One caution: turtles have strong necks and can push up a lid that is fitted loosely.
If you don't want to buy bungee cords, cut strips from either bike or car inner tubes to make homemade rubber bands. (I don't know if inner tubes are available any more.)