We present a rest-frame ultraviolet analysis of \sim 120 z \sim 3.1 Lyman Alpha Emitters ( LAEs ) in the Extended Chandra Deep Field South ( ECDF-S ) . Using Hubble Space Telescope ( HST ) images taken as part of the Galaxy Evolution From Morphology and SEDS ( GEMS ) survey , Great Observatories Origins Deep Survey ( GOODS ) , and Hubble Ultradeep Field ( HUDF ) surveys , we analyze the sizes of LAEs , as well as the spatial distribution of their components , which are defined as distinct clumps of UV-continuum emission . We set an upper limit of \sim 1 kpc ( \sim 0 \farcs 1 ) on the rms offset between the centroids of the continuum and Ly \alpha emission . The star formation rates of LAE components inferred from the rest-frame ultraviolet continuum range from \sim 0.1 M _ { \sun } yr ^ { -1 } to \sim 5 M _ { \sun } yr ^ { -1 } . A subsample of LAEs with coverage in multiple surveys ( at different imaging depths ) suggests that one needs a signal-to-noise ratio , S/N \gtrsim 30 , in order to make a robust estimate of the half-light radius of an LAE system . The majority of LAEs have observed half-light radii \lesssim 2 kpc , and LAE components typically have observed half-light radii \lesssim 1.5 kpc ( \lesssim 0 \farcs 20 ) . Although only \sim 50 % of the detected LAE components are resolved at GOODS depth , the brightest ( V \lesssim 26.3 ) are all resolved in both GOODS and GEMS . Since we find little evidence for a correlation between the rest-UV sizes and magnitudes of LAEs , the majority should be resolved in a deeper survey at the \sim 0 \farcs 05 angular resolution of HST . Most of the multi-component LAEs identified in shallow frames become connected in deeper images , suggesting that the majority of the rest-UV “ clumps ” are individual star-forming regions within a single system .