The existence of a secondary (in addition to compact object mergers) source of heavy element (r-process) nucleosynthesis, the core-collapse of rapidly rotating and highly magnetized massive stars, has been suggested by both simulations and indirect observational evidence. Here, we probe a predicted signature of r-process enrichment, a late-time (greater than or similar to 40 days post-burst) distinct red color, in observations of gamma-ray burst supernovae (GRB-SNe), which are linked to these massive star progenitors. We present optical to near-IR color measurements of four GRB-SNe at z less than or similar to 0.4, extending out to >500 days post-burst, obtained with the Hubble Space Telescope and large-aperture ground-based telescopes. Comparison of our observations to models indicates that GRBs 030329, 100316D, and 130427A are consistent with both no enrichment and producing 0.01-0.15 M-circle dot of r-process material if there is a low amount of mixing between the inner r-process ejecta and outer supernova (SN) layers. GRB 190829A is not consistent with any models with r-process enrichment >= 0.01 M-circle dot. Taken together the sample of GRB-SNe indicates color diversity at late times. Our derived yields from GRB-SNe may be underestimated due to r-process material hidden in the SN ejecta (potentially due to low mixing fractions) or the limits of current models in measuring r-process mass. We conclude with recommendations for future search strategies to observe and probe the full distribution of r-process produced by GRB-SNe.