Much as all Dwarves, including the women, are (wrongly) assumed to have long, luxuriant beards, it is commonly and (wrongly) believed that Elves cannot grow beards of their own.
While it is true that there are few bearded Elves, this is not a result of their biology. Elves are able to grow beards, but Elven fashion and tradition have historically displayed an aversion towards wearing beards, considering facial hair to be inelegant and undignified.
Even so, there have been numerous historically-notable Elves known to have worn full beards. Among these is the Elven war hero Quintus Marcellinus Styracosaurus, depicted above.