I think the answer to this lies in the etymology of the word "planet": it originally means "wanderer" in ancient Greek. From Wiktionary:
From Middle English planete, from Old French planete, from Latin planeta, planetes, from Ancient Greek πλανήτης (planḗtēs, “wanderer”) (ellipsis of πλάνητες ἀστέρες (plánētes astéres, “wandering stars”).), from Ancient Greek πλανάω (planáō, “wander about, stray”), of unknown origin. [...]
So called because they have apparent motion, unlike the "fixed" stars. Originally including also the moon and sun but not the Earth; modern scientific sense of "world that orbits a star" is from 1630s in English. The Greek word is an enlarged form of πλάνης (plánēs, “who wanders around, wanderer”), also "wandering star, planet", in medicine "unstable temperature."
And "wanderer", of course, is synonymous with "vagabond". So the phrase "to what vagabond he is a satellite himself" is merely a parallel to the previous phrase about "new satellites to Neptune": contrasting achievement in abstract scientific discovery with ignorance of the real life surrounding us. The entire passage you quote is made up of such parallels between academic study and "the art of life":
Vision through scientific instruments, but no natural vision:
"to survey the world through a telescope or a microscope"
"and never [survey the world] with his natural eye"
Knowledge of the principles of science, without its practicalities:
"to study chemistry [...] or mechanics"
"and not learn how his bread is made [or] earned"
Finding "satellites of a vagabond" in far-distant skies but not in one's own life:
"to discover new satellites to Neptune"
"and not detect [...] to what vagabond he is a satellite himself"
Observing microscopic "monsters" but not dangerous ones (here the order in the text is reversed):
"contemplating the monsters in a drop of vinegar"
"to be devoured by the monsters that swarm all around him"
The phrase you highlighted is one of these parallels. Neptune is a planet (vagabond) whose moons (satellites) might be discovered by scientific investigation, but in real life we may also be analogised to moons of a planet (satellites of a vagabond), in our orbits around important figures or ways of life. Without understanding our own lives, is it useful to understand far-off moons and abstract principles?
(More puzzling to me is why mechanics is put in parallel with how bread is earned, or why "the motes in his eyes" are included as part of the parallel with satellites of Neptune. But you asked specifically about the vagabond phrase.)