Yes, it does refer to “on“.
“Il faut ” is enough to make “on” the subject of the following verbs, you can very well start another sentence and keep it as your subject. Note that other things can make “on” the subject (c’est important…, ça serait bien…, etc.)
The possessives for “on” are “son/sa/ses“. The tonic (I think that’s the word) is “soi“, as in: “Il faut tout faire soi-même, ici !“