Yes you can, but it may vary depending on company/facility policy & procedure. Most LPN programs do not include phlebotomy or IV therapy. Many facilities offer IVT or Phlebotomy as part of their nursing education for employees. My phlebotomy training occurred on the job in a hospital setting when I worked as an ER Tech. Those that were hired after my cohort were required to have training prior to hire.
Technically in NJ no certification or credential is required to draw blood. If I recall correctly, the only states that require credentials for phlebotomists are Louisiana and California.
If you wish to take a formal class many for-profit trade schools offer phlebotomy training but these are often rather expensive. Check with your local county college (especially the community or continuing professional education department) or vocational school as they often offer phlebotomy training that is more affordable. UMDNJ offers IV therapy and phlebotomy courses for LPNs & RNs occasionally and the cost isn't too bad but priority is given to students, alumni, and current employees.