As far as I know, there were no weapon-carry laws as we have them today.
Paris is awash with weapons—halberds, pikes, poignards, pistols and muskets. Every quarter has a militia colonel who is responsible for arming his “column” in order to defend the city in case of emergency. These weapons are mostly kept in wardrobes of private homes and the storerooms of taverns, to be distributed in an emergency. But most artisans and bourgeoisie also kept something in the house with which to defend themselves. (For those without ready access to pikes and halberds, cobblestones were their chosen instrument of destruction.)
When Anne called out the French Guards and the Swiss to patrol the streets in January of 1648, the citizens voiced their displeasure by “testing” their muskets all night—within earshot of the Palais Royal.
Sword nobles had the right to carry a sword in the defense of the realm. It was a special privilege that they jealously guarded.
And there is a law on the books that prohibits armies of any type—including French!—from coming within twenty miles (or so) of Paris.
Lastly, Paris of 1648 is the royal capital but also a city with independent rights, as governed by the Provost of Merchants and Parlement. This dynamic creates an uneasy detente between civic and royal power. And the Provost is well aware that angering the Queen is never in his best interest. Therefore, while there are no police per se, the Provost and Parlement take law and order very seriously. In particular, the bourgeoisie detest “vagabonds:” Beggars, soldiers, thieves and anyone else who looks like they might form a crowd and cause a disturbance. The Provost will gather a force of armed bourgeoisie and drive out any such group of vagabonds that he should find within the city’s precincts.
So your friend is in a tricky place. Is he a heavily armed, dangerous looking vagabond? Then the authorities will likely seek to arrest him. Is he a well dressed, well armed soldier of fortune looking for work? Then he’ll likely find patrons inquiring after his services. It all depends how he carries it.
Now it’s also important to note that entry to Paris was tightly controlled by Impost Gates—customs agents that searched for contraband being brought into the city. They levied fines and taxes on all goods entering the city, and impounded anything illegal (mostly banned books, afaik). Unless there is some intrigue, these agents will not allow a wagonload of weapons to be brought into the city by just anyone.
So what is appropriate for your soldier of fortune on the city streets? A rapier, a pistol and a cloak are de rigeur. Heavier hardware can be stashed at his accommodations.
As for ye olde weapon shoppes, absolutely not. Gunsmiths, probably a few who can make custom arms or do repairs. But again, the city is lousy with weapons, so I wouldn’t make acquiring a musket a conflict. You just have to know who to ask.