We present optical, radio, and X-ray observations of SN 2020bvc (=ASASSN-20bs, ZTF 20aalxlis), a nearby broad-line (BL) Type Ic supernova (SN) and the first double-peaked Ic-BL discovered without a gamma-ray burst (GRB) trigger. Our observations show that SN 2020bvc shares several properties in common with the Ic-BL SN 2006aj, which was associated with the low-luminosity gamma-ray burst (LLGRB) 060218. First, the 10 GHz radio luminosity is brighter than ordinary core-collapse SNe but fainter than LLGRB SNe such as SN 1998bw (associated with LLGRB 980425). We model our VLA observations (spanning 13-43 days) as synchrotron emission from a mildly relativistic (v 0.3c) forward shock. Second, with Swift and Chandra, we detect X-ray emission (L X 1041 erg that is not naturally explained as inverse Compton emission or part of the same synchrotron spectrum as the radio emission. Third, high-cadence (6× night-1) data from the Zwicky Transient Facility (ZTF) show a double-peaked optical light curve, the first peak from shock cooling of extended low-mass material (mass SN 2020bvc is the first double-peaked Ic-BL SN discovered without a GRB trigger, so it is noteworthy that it shows X-ray and radio emission similar to LLGRB SNe. For four of the five other nearby (z ≲ 0.05) Ic-BL SNe with ZTF high-cadence data, we rule out a first peak like that seen in SN 2006aj and SN 2020bvc, i.e., that lasts 1 day and reaches a peak luminosity M -18. Follow-up X-ray and radio observations of Ic-BL SNe with well-sampled early optical light curves will establish whether double-peaked optical light curves are indeed predictive of LLGRB-like X-ray and radio emission.