It would be not only okay, but wildly applauded if you managed to capture Henry V personally on the fields of Agincourt. Typically, your hope and wildest ambition would be to capture him rather than kill him, but no one would hold it against you if you happened to put a poleaxe through his skull. A live king in your hands is a lucrative financial opportunity for you, a legendary capstone for your career and reputation as a soldier, and a massive political coup for the kingdom of France. They won't pay much of a ransom for a dead king. You may be one of a group of 18 French knights and men-at-arms who swore to do exactly what you have in mind at Agincourt and were praised for their valor by contemporary chroniclers and eyewitnesses. Not one member of this group would survive the fight.

Your English counterparts are as determined to fight for their king (and preserve their own lives!) as you are to defend France from what you see as a foreign usurper. Several English men-at-arms will go on to earn knighthoods, glory, and financial windfalls from capturing well-known French aristocrats later in the day- living the dream that you had imagined for yourself! Both sides, contrary to the mythological vision of the arrogant, upper-crust French against humble, salt-of-the-earth English, are predominately made up of professional or semi-professional contract soldiers and retainers. Both you and the English soldiers who will probably leave you dead in the mud here on this battlefield are extremely aware of the consequences of defeat. Pitched battles in the field between such large forces were not common and even lifelong veterans may have only fought in a handful of them throughout their entire career. The stakes are high, literally and figuratively, at Agincourt as you watch English archers down the field set up their defenses against a cavalry charge.

The main problem with the plan of attacking the king directly was not any rules of engagement that prohibited attacking monarchs directly, but the qualities of the monarch in question (and of his army). Henry V was not the inexperienced and unsure figure depicted in Shakespeare, but a hardened, experienced soldier who had previous survived a deep arrow wound to the face. He fought personally at Agincourt and was reported to have personally defended his younger brother Humphrey who was lying wounded in the field. The chronicler Enguerran Monstrelet claims that the Duke of Alencon got close enough to actually knock off a piece of Henry's crown with an axe, but then found himself isolated and surrounded by English troops. As the Duke attempted to surrender, he was slaughtered by Henry's enraged bodyguard. If their fury at watching an attack on their king drove them to kill such a potentially lucrative prisoner, a French soldier attacking Henry V who was of far lesser status (i.e., you) could probably expect a similar lack of mercy. One French knight was put into a nearby farmer's house for safekeeping after he was wounded and captured during the battle. When Henry V ordered the prisoners to be slaughtered, the house was set on fire, killing the majority of prisoners inside.

The Duke of Alencon's story illustrates the fact that it was not taboo whatsoever to take a (potentially fatal) swing at a king. If a king was fighting on the field, he was as fair a target as any other soldier. The fact that a group of 18 men specifically attempted to take out Henry personally (as well as the Duke's attempt) shows that kings were viewed as priority targets, whose death or capture could potentially turn the tide of battle. However, kings knew this well themselves, and had personal bodyguards to fight alongside them and defend them. Henry's presence in the frontlines was claimed in English sources to be a significant boost to English morale at Agincourt, and while this is obviously mentioned for purposes of propaganda, the English army's ability to remain cohesive and effective throughout an extremely brutal and hard-fought pitched engagement does suggest that their spirits were high. Henry took a calculated risk and it paid off enormously well for him, sealing his reputation as a warrior king for centuries to come.

This thread about commoners killing royalty in battle is similar, and u/MI13 actually touches on other topics like common soldiers capturing knights.

Incidentally, "man-at-arms" is often used to mean "common soldier" but this is anachronistic. In the example of the 18 knights and men-at-arms cited above the men-at-arms would have been of lower social status than the knights but not by much and would probably have come from the gentry, albeit maybe younger sons or cadet branches, unless they were common soldiers who had worked their way up. Men-at-arms (g'endarmes in French) were gentlemen entitled to bear arms, which means both that they had the means to equip themselves with full armour and at least a couple of horses plus one or two servants (varlets) to look after the horses and equipment ("gros varlets" would fight and have weapons of their own, and resemble what you might have had in mind as a "man-at-arms"); but they could also "bear arms" in the sense of being entitled to display heraldic arms ie what tends now to be called a "coat of arms"