Form 15 G & H

Yes, a society or trust is eligible to submit Form 15G, if it satisfies the requisite conditions. But Form 15H is only for resident senior citizens.
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Sec 197(1A): Notwithstanding anything contained in section 192A or section 193 or section 194A or section 194DA or section 194K, no deduction of tax shall be made under any of the said sections in the case of a person (not being a company or a firm), if such person furnishes to the person responsible for paying any income of the nature referred to in section 192A or section 193 or section 194A or section 194DA or section 194K, as the case may be, a declaration in writing in duplicate in the prescribed form and verified in the prescribed manner to the effect that the tax on his estimated total income of the previous year in which such income is to be included in computing his total income will be nil.