§73.816(a) of the rules states:
Permittees and licensees may employ nondirectional antennas with horizontal only polarization, vertical only polarization, circular polarization or elliptical polarization.
Full power stations are permitted to operate both horizontal and vertical antennas at the same time. There are fields in the FCC's CDBS database to accomodate such operations.
Since §73.816(a) states that LPFM stations can run horizontal only and vertical only polarization, they are not permitted to run combined antennas thus only needing one of the two fields.
So yes, you can run a vertical antenna as long as it is not in conjunction with a horizontal antenna.
We have received some feedback questioning whether circularly polarized antennas such as the "penetrator" style antennas are OK on LPFM. The answer is yes, they are.