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The Estonian Consulate in New York explains what documents are acceptable:
applicants who arenβt US citizens, but reside in US: valid US residence permit or resident visa type A, E, F, G, H, I, J, L, O, R. F visa holders must present together with visa also the form I-20 with valid travel endors*m*nt. J visa holders must additionally present form DS-2019 (J1) with valid travel endors*m*nt. Attention! Green card, visa and additional forms must be presented in original together with a copy.