Basic drawing is now available in Day One iOS starting in version 3.4. You can use your finger or Apple Pencil (iPad only).
Drawing is a Premium feature.
Tap to start a new blank entry, then tap the paper clip icon above the keyboard and choose Draw.
Tap Done at the top right to save the drawing.
To edit a drawing, tap the drawing in the entry, then tap Edit. This will allow you to add to the drawing, but original edits cannot be edited.