We discuss new observational constraints on the abundance of faint high redshift Lyman-alpha emitters secured from a deep Keck near-infrared spectroscopic survey which utilizes the strong magnification provided by lensing galaxy clusters at intermediate redshift (z=0.2-0.5). In each of 9 clusters we have undertaken a systematic `blind' search for line emission with NIRSPEC in the J-band within carefully-selected regions which offer very high magnifications (>10-50x) for background sources with redshifts z~10. The high magnification enables the detection of emission at unprecedented limits (10^41 - 10^42 erg/s), much fainter than those of conventional narrow band imaging and other spectroscopic searches. As the comoving volumes probed are small, our survey is designed to address the important question of whether intrinsically feeble star forming galaxies could provide the dominant ionizing flux at z~10. Our survey has yielded six promising (>5 sigma) candidate Lyman-alpha emitters which lie between z=8.7 and z=10.2. While the predicted reionization photon budget depends upon a large number of physical assumptions, our first glimpse at the z~10 Universe suggests that low luminosity star-forming galaxies contribute a significant proportion of the UV photons necessary for cosmic reionization. [Abridged]