Only five US presidents have sported full beards, and another four had moustaches of varying degrees of glory. These were all during the half century run of the dozen Presidents between Lincoln and Taft, of whom only Andrew Johnson and McKinley were clean-shaven.
The Republicans have historically been the hairier party. Every Republican candidate from 1856 to 1892 (with the arguable exception of 1860) had a beard, with a later run of moustaches on Roosevelt in 1904, Taft in 1908 and Roosevelt and Taft in 1912 followed by a final beard in 1916 (and a postscript for the moustache in 1944 and 1948). We should not forget also the vast mutton-chop whiskers of Chester A. Arthur, who was elected Vice-President on the Republican ticket in 1880 but served most of that term as President after the assassination of the bearded Garfield.
The Democrats have never had a properly bearded candidate; the only successful Democrat who even went as far as a moustache was Grover Cleveland, who won the popular vote three times running in 1884, 1888 and 1892. (Cleveland lost the electoral college in 1888 to Benjamin Harrison, so far the last American President with a proper beard.) The hairiest Democrat was Winfield Scott Hancock, whose huge moustache did not help him in the 1880 election, and Democrats with moustaches lost in 1864 and 1904.