Nowhere is the adhesion of soft materials as fascinating than it is when biological cells adhere. In the case of endocytosis, infection of the cell by viruses, and uptake of certain types of delivery particles, not only does the surface of a cell adhere a target, the cell membrane and nearby cyctoskeleton rearrange to engulf that object. This process involves bending of the membrane, which comes with an energetic cost, distinguishing the behavior from classical wetting, where there is an energy cost to make new surface, but none to bend the interface. This work explores the extent to which adhesion and engulfment can be driven at the level of the membrane, and the role of bending energy in producing kinetic signatures for engulfment behavior. Polymer vesicles with different bending moduli and different adhesive driving forces access different regimes of variable space, demonstrating when engulfment is rapid, when it follows a lag time, and when it is altogether unfavorable.