Don't want to discourage new answers of course, but I answered a similar question a couple of years ago. Since the user deleted their question and their account, I guess I can repost my answer here:

There’s a scene in Lawrence of Arabia where people crowd around the new governors of Damascus waving their written petitions in the air, hoping to be heard - that’s not medieval, but I think that’s pretty much how people picture a royal court in the Middle Ages. Well, it probably wasn’t very much like that, but similar things did happen, sometimes. I’m sure there are an infinite number of examples we could give from all over the medieval world, but I can describe how it worked in France and England around the 12th and 13th centuries.

“All medieval kings considered it their foremost duty to offer justice through their personal court, the curia regis” (Baldwin, p. 37)

How did they offer justice? Monarchs usually didn’t have a single court in a single place. The king moved around a lot, so the “court” could be at whatever castle or palace or town he happened to be in. In England, after the Norman Conquest the government was pretty centralized, and the king had the authority and (more importantly) the power to enforce the laws throughout the whole kingdom. The justice system developed so that royal officials travelled around the country, listened to people’s complaints, and rendered decisions on behalf of the king. If you had a complaint or wanted to accuse someone of a crime, you’d have to wait until the judges showed up. But if you lived in London, or you were wherever else the king was, you might be able to appeal to the king himself.

In France, the government was much less organized. Up until the 13th century, the kings had very little authority and virtually no power outside Paris and the immediate area (basically the modern Île-de-France), so they could hardly send royal officials to dispense justice throughout the entire kingdom. But within that relatively small territory, people could bring their complaints to the king directly, wherever the king might be - not necessarily in Paris, but maybe Fontainebleau, Melun, Bourges, Senlis, etc etc…and also in Normandy, once the French took Normandy back from the English.

Conquering Normandy was really the first step in making France a more centralized kingdom governed from a capital, like England was. But it was a very gradual process and the legal system developed slowly.

“No fundamental changes were introduced in the operations of the curia regis until well into the reign of Louis IX, when the time and place of sessions were finally scheduled, a permanent corps of judges was created, and records appeared in the famous Olim of 1248.” (Baldwin, pg. 43)

Louis IX established the “Palais de Justice” in Paris so everyone would have one central place where they could hear their cases. That didn’t necessarily mean he himself would be sitting there hearing cases, but now there were judges and a centralized legal system, like England had. However, he was known for his attention to legal matters, and apparently he did like to hear cases himself:

“During the summer he often went and sat in the woods of Vincennes after Mass. He would lean against an oak tree and have us sit down around him. All those who had matters to be dealt with came and talked to him, without the interference of the ushers or anyone else. He himself would ask ‘Is there anyone here with a case to settle?’ Those who did have a case stood up, and he said to them, ‘Everyone be quiet and you will be given judgement, one after another’…Whenever he thought something said by those who spoke on his behalf or on another’s behalf needed correction, he said himself how it should be amended. On some summer days I saw him go to the gardens in Paris to render justice to the people...He had carpets laid out so that we could sit round him. Everyone who had a case to bring before him would gather around him at first, and then the king had judgement delivered in the same way as I said took place in the wood at Vincennes.” (Joinville, pg. 157)

This specific story, I think, is often what people are thinking of when they imagine the scenario in your question. Stories like this, from Jean de Joinville and other medieval authors, were used in Louis’ favour when the church was investigating whether or not he should be declared a saint. So, they probably fall more into the category of hagiography than biography - here Joinville is probably alluding to King David or King Solomon from the Bible, who also heard complaints and dispensed justice this way. Did Louis really sit on a carpet under a tree? Maybe not…but he definitely did reorganize the legal system in France centralize the government.

“Louis and his contemporaries did not doubt that the primary duty of a king was justice” and Louis was eager “to give exact justice to all, rich and poor, high and low.” (Labarge, pg. 169)

But who were these people? Did peasants and “commoners” really have direct access to the king’s court? The records of Louis’ early courts are kind of sparse, and before Louis there are hardly any records at all, but it looks like they court mostly dealt with disputes among the nobles and among the church. A peasant probably wouldn’t be attending the royal court. First of all, peasants by definition didn’t live in the city; there would be merchants and tradespeople there, who were “commoners” in the sense that they weren’t aristocrats, and they probably had access to the court, but rural peasants out in the countryside probably didn’t.

In France and England, the rural peasants were usually serfs, so they were tied to the land they lived on and didn’t have much freedom of movement, and they were sort of considered the personal property of whichever local lord owned that land. If they had legal disputes, their most immediate legal authority was that local lord. In a centralized kingdom like England and, eventually, France, the king’s justice applied to the poorest serf as well as the richest noble, but the local nobility was responsible for administering that justice.

So, in short, yes, people did sometimes appeal directly to the king to settle their legal problems, and some kings were well-known for their interest in legal disputes. But the people who came to thee king were probably nobles or clergy, not peasants.

Hopefully other experts can talk more about other places and eras - Charlemagne was also supposed to have heard legal cases from his subjects, but that’s a bit too early for me.

Sources:

Richard Mortimer, Angevin England, 1154-1258 (Blackwell, 1994)

John W. Baldwin, The Government of Philip Augustus: Foundations of French Royal Power in the Middle Ages (University of California Press, 1991)

Margaret Wade Labarge, Saint Louis: Louis IX, Most Christian King of France (Little Brown and Company, 1968)

Joinville and Villehardouin: Chronicles of the Crusades, trans. Caroline Smith (Penguin, 2009)