One of the open questions following the discovery of GW170817 is whether neutron star mergers are the only astrophysical sites capable of producing $r$-process elements. Simulations have shown that 0.01-0.1M$_⊙$ of $r$-process material could be generated in the outflows originating from the accretion disk surrounding the rapidly rotating black hole that forms as a remnant to both neutron star mergers and collapsing massive stars associated with long-duration gamma-ray bursts (collapsars). The hallmark signature of $r$-process nucleosynthesis in the binary neutron star merger GW170817 was its long-lasting near-infrared emission, thus motivating a systematic photometric study of the light curves of broadlined stripped-envelope (Ic-BL) supernovae (SNe) associated with collapsars. We present the first systematic study of 25 SNe Ic-BL -- including 18 observed with the Zwicky Transient Facility and 7 from the literature -- in the optical/near-infrared bands to determine what quantity of $r$-process material, if any, is synthesized in these explosions. Using semi-analytic models designed to account for $r$-process production in SNe Ic-BL, we perform light curve fitting to derive constraints on the $r$-process mass for these SNe. We also perform independent light curve fits to models without $r$-process. We find that the $r$-process-free models are a better fit to the light curves of the objects in our sample. Thus we find no compelling evidence of $r$-process enrichment in any of our objects. Further high-cadence infrared photometric studies and nebular spectroscopic analysis would be sensitive to smaller quantities of $r$-process ejecta mass or indicate whether all collapsars are completely devoid of $r$-process nucleosynthesis.