Beards were distinctly out of fashion for anybody but beatniks, college professors, and other groups outside Middle America for pretty much all of Nixon's career in politics. I can't see one helping with his intentionally square image - it might harm his electoral prospects.
Assume that Nixon, for whatever reason, decides to grow a beard during his first term as VP. This puts off some people, including some Republican honchos, and even Ike. But Nixon is not dropped from the ticket in 1956, because no one wants to say publicly, or even privately, that something as important as the Vice Presidency should be controlled by anything as trivial as whiskers.
By 1960, Nixon's beard is an accepted institution. And it finally has an effect: in the TV debate with Kennedy, bearded Nixon comes across better than 5 o'clock shadow Nixon did OTL. This causes a small shift in voting, about 0.5%, which is enough to flip Hawaii, Illinois, Missouri, New Jersey, and New Mexico, with 62 EV. That gives Nixon a majority of 281.