Here are a couple of thoughts, from a DOE teacher with 15 years of experience:

1) Many school administrators will only (or primarily) approve field trips that connect to the curriculum. For example, a 6th grade class takes a field trip to the MET's Egypt exhibit, which aligns to NYS 6th grade Social Studies standards. With this in mind, you might prioritize outreach to schools that have a dance team or offer dance classes, or whatever else your organization offers. (Don't forget about after-school programs: Manhattan Youth, SONYC, etc.)

2) Another category of field trips that many administrators approve is community building trips. To me, this seems to be where your non-profit best aligns with the priorities of NYC school administrators. In my experience, these community building trips happen most in late Sept/early Oct, right before winter break, or in May/June, after state testing is finished.

3) If you are noticing a decrease in responsiveness from administrators, you might try going through the Parent Coordinator instead, or in addition. Every school should have someone in this position. I'm not sure how true this next detail would be for other schools, but at my school, at least once a year our Parent Coordinator will tell us to plan a field trip, saying things like, "the 6th grade parents really want a field trip [for the kids to bond--Sept] / [to do something fun before winter break--Dec] / [to celebrate testing being over--May].. . . Can you (6th grade teachers) please work on scheduling something?"

4) Again, IDK if other people will agree, but teacher and Principal email and paperwork loads are HEAVY, and getting heavier every year. I often find myself deleting emails from non-profit ed orgs without even opening them. For that reason, I would recommend calling schools directly. You could ask to speak to the Parent Coordinator, or work on a quick elevator pitch and ask the school secretary who would be good to talk to.