This answer seems to imply that it was a longbow.
Is there canon information on what kind of bow would Elves use? (and if so, was it different for different Elvish groups?)
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There is no one single type.
The Teleri in Valinor are described in the Silmarillion chapter Of the Flight of the Noldor:
For the Noldor were become fierce and desperate, and the Teleri had less strength, and were armed for the most part but with slender bows.
Beleg Strongbow has something a bit more substantial (from Of Túrin Turambar):
...they laid Beleg in a shallow grave, and placed beside him Belthronding his great bow, that was made of black yew-wood.
The difference between the bows of Mirkwood Elves and those of Lórien Elves is described at Galadriel's gift-giving in Lord of the Rings:
To Legolas she gave a bow such as the Galadhrim used, longer and stouter than the bows of Mirkwood, and strung with a string of elf-hair.
But in general Tolkien doesn't really describe this kind of level of detail, except where it's important to the plot. There are other references where only possession of a bow is mentioned without specification of type: "Fingon strung an arrow, and bent his bow", "Curufin, being filled with shame and malice, took the bow of Celegorm".