When was the word "ambigu" first used with the sense of "meal with all items served at the same time"?
The word ambigu used as a noun meaning Repas où l'on sert à la fois les viandes et le dessert can be found before the date mentioned in the question (1751) since we find it in the Mémoire de la vie du comte de Gramont that was written by Antoine Hamilton between 1704 and 1710 and first published in 1713.
Here's the sentence as found in a 19th century reprint:
Souvent c'étoient des ambigus qui partoient aussi de France pour enchérir au milieu de Londres sur les collations du roi.
Hamilton also used the word in another writing that can be found in his complete works:
le couvert avoir été mis dès le grand matin, au jeu de boule; la symmétrie fut dérangée par la précipitation dont on déménagea ; quelques pièces de l'ambigu se perdirent en chemin ; on servit tout de travers, & le vin manqua ;
Those are the earliest occurrences I have found so far, but the word must have been used in French before the 18th century since we find it used in English in the 17th century and the OED says it is borrowed from French. The definition given by the OED is: "An entertainment at which the viands and dessert are served together; or at which a medley of dishes are set on" and there is a quote from the London Gazette dated from 1688:
They were all entertain’d to their Satisfaction, at a very splendid Ambigu. (The London Gazette, 1688)
Is the word ambigu still used in French or has it been replaced by buffet or any other word?
The definition of ambigu used as a culinary term is:
Repas froid où l'on servait à la fois tous les mets et dessert (Grand Larousse).
We note the past tense (on servait), indeed the word isn't used anymore and buffet is the usual word for that type of thing, even if a buffet is not necessarily cold, it often is.
I suppose that a meal with the main courses and the desserts served at the same time can be a bit ambiguous.
I would rather relate ambigu here to its meaning of mélange rather than directly to the adjective ambigu. It is the same use of the noun ambigu as found for a hybrid mix of music:
ariettes, ambigus ou romances tirées des opéras-comiques en vogue (Sainte-Beuve, Les causeries du Lundi)
or in the name of the Théâtre de l'Ambigu-Comique, name given because of the wide variety and mix of theatrical modes staged there.