There were plenty of people that lived outside the feudal system in Medieval Europe. First of all, there were countries that were never feudal, such as Norway, Sweden and Finland (the latter being integrated into Sweden during the high medieval period, step by step). Then there were places such as Dietmarschen, Gotland and Jämtland that were effectively peasant republics and while they may nominally belong to a realm or a ruler, they had no duties towards that realm or ruler, and effectively ruled themselves.

Then you have various city states and merchant republics such as Florence, Venice, Genoa and Ragusa, which may control land held in a feudal system in their hinterlands they themselves were not feudal states.

Then you have free towns and cities with city charters, where in exchange for a fixed fee or part of the income or tolls of a town, a lord or monarch relinquished his or her rights within the town perimeter, and local laws, upholding them, the maintenance and defence of the walls and so on was handled by the burghers of that town or city. Some of these became rich and powerful enough to be military powers of their own, such as Lübeck, one of the most powerful cities of the Hansaetic League in northern Germany.

Aside these places that were not feudal states, there were people that made their living travelling. Moneylenders, merchants of various kinds, tailors, carpenters, seamstresses, smiths, jewelers and others that could be travelling from fair to fair or to seasonal work. Skilled labour, such as master masons, sculptors, artists, painters, quarrymen and the less skilled labour associated with them (such as toolsmiths, oxcarters and other hard labour) often travelled between projects such as church and cathedral construction, castle building or renovations or expansions of larger buildings.

In general, the church attempted to keep tabs on the population for census prurposes and making sure everyone attended services regularly and did not suddenly get any heretic ideas. They also kept tabs on land onwership for tithing purposes. Feudal lords, and in non-feudal places those that collected the taxes for the state usually kept tabs on land ownership and who owed taxes (generally either in form of corvee labour or in form of natural goods in this era). The burgher councils or similar in the cities kept tabs on who owned property or had a place in a guild (or similar organisation) and thus the right to conduct business within the town or city for tax purposes.

But otherwe, there were groups that made their living trabelling, or working for some time in one place and then going to the next. They ranged from mercenaries to highly sought skilled artisans to farm hands and seasonal labour, to travelling merchants going from fair to fair, to camp followers, prostitues, gamblers, mobile innkeepers and merchants making up the tail of armies.