No. That is considered "premature construction". Premature construction is prohibited pursuant to Section 319(a) of the Communications Act.
However, the FCC does tolerate certain activities prior to a construction permit grant, including:
- Site clearance,
 - Pouring of concrete footings for a tower,
 - Installation of a tower base and anchors,
 - Installation of a new power line,
 - Purchase and on-site storage (but not installation) of radio equipment, and
 - other preliminary steps having no intrinsic broadcast use.
 
Pre-authorization construction of towers or installation of radio antennas, however, is strictly prohibited.
LPFM enhancements and change applications
          LPFM station planning and construction
          LPFM during the filing window
              
    