Everyone wondered how the travel vlogger could go around the world all through the year and ______.

The sentence expresses curiosity about how a travel vlogger manages to sustain constant worldwide travel.The blank needs a phrase that accurately captures the essence of his perpetual movement and how he handles it. 

The best answer is "manage his itinerant lifestyle" because "itinerant" precisely describes a travelling way of life. This fits perfectly as it's the correct grammatical form (adjective + noun) and matches the travel context. Other options fail because: "itinerary" is a noun (not an adjective), "iterative" means repetitive (unrelated to travel), and "itinerary in his lifestyle" is grammatically awkward. Only "itinerant" properly conveys the vlogger's mobile existence.