There appears to be a FedEx drop box on the second floor of JFK T4. (Whether it is airside or landside I do not know.) I would call FedEx to confirm. You should be able to go airside at T4 -- if necessary -- with a same-day boarding pass for a flight departing from any JFK terminal.

Both UPS and FedEx drop boxes have room to hold supplies: next-day and second-day air envelopes, shipping labels, etc. But I would never rely on any particular box having the supplies I need on any particular day. If you can obtain what you need from a FedEx drop box or office before you arrive at JFK, that would be prudent. Depending on the package's destination, FedEx's cheaper Ground Service might be all you need for timely delivery.