Is it permissible to continue distributing stickers that contain the ذكر الله (remembrance of Allah), knowing that they may fall to the ground, and would that entail a sin?

There is no harm in stickers that bear reminders of Allah and call to goodness and good manners, for they are a form of calling to goodness. They must be placed in respectful locations where they will not be subjected to disrespect. If they fall unintentionally, there is no sin.