Can I ask if other applicants want the apartment?

I applied for a lease and asked the property manager if there were other applicants for the unit I'm looking at. She declined to answer my question. Can she do that? Yes. She can decline to answer the question if the landlord has instructed her to do so, because property managers are not required to inform you about other applicants. Keep in mind that if you submitted a TAR Residential Lease Application, that form states that the landlord can continue to show the property to other prospective tenants and can accept another offer absent a written agreement otherwise.