There is nothing in NYS law to directly prevent shipping of ammo.
There *is* an ammo background background check written into SAFE that would be impossible to comply with in anything other than a face to face transaction.
However, that background check requirement dosen't take effect until the NYSP implement the required infrastructure to support it. They have not done so.
So, as long as you can still buy retail ammo without a background check, you can still have it shipped.
Finding a business that wants to deal with this is much harder. The vast majority just stopped shipping as soon as the law was passed under the assumption that the database would be quickly implemented.
There are a few good ones out there. The search function should reveal them. If not PM might be a better bet.
People are reluctant to advertise "their guy" publicly for fear that FUAC's goon squad will harass them and they will stop. There is plenty of precedent for this behavior.