We observe two metal-poor main-sequence stars that are members of the recently discovered Sylgr stellar stream. We present radial velocities, stellar parameters, and abundances for 13 elements derived from high-resolution optical spectra collected using the Magellan Inamori Kyocera Echelle spectrograph. The two stars have identical compositions (within 0.13 dex or 1.2{sigma}) among all elements detected. Both stars are very metal-poor ([Fe/H]=-2.92+/-0.06). Neither star is highly enhanced in C ([C/Fe]<+1.0). Both stars are enhanced in the {alpha} elements Mg, Si, and Ca ([{alpha}/Fe]=+0.32+/-0.06), and the ratios among Na, Al, and all Fe-group elements are typical for other stars in the halo and ultra-faint and dwarf spheroidal galaxies at this metallicity. Sr is mildly enhanced ([Sr/Fe]=+0.22+/-0.11), but Ba is not enhanced ([Ba/Fe]{<}-0.4), indicating that these stars do not contain high levels of neutron-capture elements. The Li abundances match those found in metal-poor unevolved field stars and globular clusters (GCs) (log{epsilon}(Li)=2.05+/-0.07), which implies that environment is not a dominant factor in determining the Li content of metal-poor stars. The chemical compositions of these two stars cannot distinguish whether the progenitor of the Sylgr stream was a dwarf galaxy or a GC. If the progenitor was a dwarf galaxy, the stream may originate from a dense region such as a nuclear star cluster. If the progenitor was a GC, it would be the most metal-poor GC known.