|Hussar of Lauzun's legion (uniformology.com)|
In 1778, Armand Louis de Gontaut, duc de Lauzun, was authorised by the Minister of Marine (who handled all French overseas colonies, including their defenses) to raise a corps of eight legions to supplement existing naval and military forces for combat around the world. ("Legion" in this sense was an eighteenth century term for formations that combined infantry, cavalry, and sometimes artillery, like mini-armies.) These Volontaires Étrangers de la Marine would be recruited not from French subjects but from foreigners (étrangers)--in this case mostly Germans, Irishmen, and Poles--a fairly common practice in the French military (the French army had many regiments composed at least notionally of non-French troops, including Germans, Swiss, Irish, Scots, and Swedes).
The notional establishment of each of these legions was to be eight companies: one grenadier (~100 men), one chasseur (~170), two fusilier (~170 each), one artillery (~170), two hussar (~170 men each), and one artificier/engineer.
The VEM corps also had a headquarters company (Compagnie Generale) of about 100 officers and hussars.
One legion was sent off to West Africa and then served in the Indian Ocean and southeast India. Another served in the Caribbean. The 2me VEM was originally slated for an invasion of the UK. This invasion never took place, so the legion was thus available when the French government needed more troops, especially light, irregular troops to send to America.
Amalgamated with the corps' Compagnie Generale and the Volontaires Étrangers de Nassau (another foreign regiment that Lauzun had been given to command), the 2me VEM became known as the Volontaires Étrangers de Lauzun or the Légion de Lauzun.
A problem emerged when it came time to ship the Legion overseas; the Navy had not enough transport to carry the whole force. In consequence, many of the hussars' horses, some of the grenadiers, chasseurs, and artillerists, and all (!) of the fusiliers were left behind. It would appear that the fusiliers never joined the Legion in America and were instead engaged in operations in Europe.
The strength of the two hussar squadrons as shipped were 159 and 136 respectively, but they took with them only about 170 horses. They purchased more horses after arriving in America.
The grenadiers' strength in America (troops shipped plus drafts from *French* units) was at least 127 men; the chasseurs' was at least 111. The artillery had at least 110 men with four 4-pound guns.
I say "at least" because in addition to their drafts from other French units, the Legion seems to have recruited about 60 more men after they landed in America and before arriving at Yorktown. But these also seem to have been largely German POWs and deserters held by the Americans, and the unit suffered heavily from desertion (very possibly from these same men), so it's hard to know exactly what its strength was after these additions and subtractions.
In America the Legion was also joined by one hussar squadron from the 1me Volontaires Étrangers de la Marine and a detachment from the Regiment d'Infanterie Dillon (the commander of the Legion's cavalry was a Dillon, so the detachment of infantry from that regiment may or may not be a coincidence). It's not clear how large these attachments were, but between the addition of the 1me hussars and purchase of horses in America, the mounted strength of the Legion seems to have been 250 at this point and the strength of the two infantry companies about the same.
The Legion was engaged in two skirmishes while in America; one in New York and the more well known one in Virginia. Reading French accounts of these engagements and comparing them to American and British accounts is an entertaining endeavour, as well as a good exercise in comparing differing historical sources. :-) Let's just say that the French are never behindhand in describing their own valour.
JD Glasco's Gentlemanly Wargaming includes this blogpost about the Legion, which give some detailed organizational notes, including both paer strength and actual troops carried from France and recruited in America. Most of what I report above is based on his research.
A reenactment group who portray the Legion have this webpage on its history.
The JAR has this article on the Legion.
Uniformology has a brief history of the unit online, as well as descriptions of uniforms and a colour plate of one of the hussars.
This page lists the service of all French units during the war and mentions that Lauzun's unit was at Yorktown, as were the hussars of the 1me Legion, but that the 2me's fusiliers served instead in the Netherlands in 1782.
americanrevolution.org has Robert Selig's chatty but somewhat scattershot The Duc de Lauzun and His Legion
Although it doesn't contain all the details one might want, it's worth noting that the memoires of the Duc himself is available free on Googlebooks.