You were refused. Find out which jurisdiction said no.
A New York refusal has two very different possible meanings, and they need different responses. Either you did not meet the statutory test — which you can grieve — or your taxing jurisdiction simply never adopted the exemption, which is lawful and cannot be grieved. Establish which one you are dealing with before doing anything else.
N.Y. Real Prop. Tax Law §458-a; §524 (grievance)
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Your grievance route
If the refusal was a finding against you, file a grievance with the Board of Assessment Review. It sits on the fourth Tuesday in May in most towns, though cities and villages differ. If instead your jurisdiction never adopted the exemption, there is nothing to grieve — the route there is political rather than legal, and your county veterans service agency will know whether an adoption is under consideration.
New York does not prescribe what a denial notice must contain the way some states do, so there is no "the letter is defective on its face" argument here. What New York has instead is Grievance Day and a genuine ambiguity worth resolving first: was this a finding against you, or a jurisdiction that never offered the exemption?
One more thing worth checking
The most common confusion in New York is not a wrong denial at all — it is a veteran concluding they are ineligible when in fact their town simply never adopted the exemption, while the neighbouring town did. The second most common is a veteran refused the Alternative exemption for lack of wartime service who is never told the Cold War exemption exists for exactly that gap.
Read the record on the Cold War exemption