Our ability to design artificial micro/nanomachines able to perform sophisticated tasks crucially depends on our understanding of their interaction with biosystems and their compatibility with the biological environment. Here, we design Janus colloids fuelled only by glucose and light, which can autonomously interact with cell-like compartments and trigger endocytosis. We evidence the crucial role played by the far-field hydrodynamic interaction arising from the puller/pusher swimming mode and adhesion. We show that a large contact time between the active particle and the lipid membrane is required to observe the engulfment of a particle inside a floppy giant lipid vesicle. Active Janus colloids showing relatively small velocities and a puller type swimming mode are able to target giant vesicles, deform their membranes and subsequently get stably engulfed. An instability arising from the unbound membrane segment is responsible for the transition between partial and complete stable engulfment. These experiments shed light on the physical criteria required for autonomous active particle engulfment in giant vesicles, which can serve as general principles in disciplines ranging from drug delivery and microbial infection to nanomedicine.