In my proof that mapping class groups are automatic, Ann. of Math. (2) 142 (1995), no. 2, 303–384, I used a theorem from ECHLPT "Word Processing in Groups" which says that if a groupoid is automatic then the corresponding group is automatic.
That theorem was applied in the situation of a finite type surface $S$ with one or more punctures, using the groupoid mentioned in Bruno Martelli's answer which has come to be called the "Ptolemy groupoid" of $S$, due to connections with work of Robert Penner. That groupoid needs to be altered slightly for purposes of my proof, by adding data which breaks the finite symmetry group of an ideal triangulation. The data I added was an enumeration of the prongs of the triangulation, so the objects of the resulting groupoid are "ideal triangulations with enumerated prongs". The generating morphisms of this groupoid are of two types: permutations of the enumeration; and the flip relators mentioned by Bruno Martelli, called "elementary moves" in my paper, together with some rule for enumerating the prongs of the new ideal triangulation resulting from the elementary move.
The group corresponding to this groupoid turns out to be the mapping class group of $S$, and hence the theorem from ECHLPT is applicable.