We estimate the 21 cm radio background from accretion onto the first intermediate-mass black holes between z ≈ 30 and z ≈ 16. Combining potentially optimistic, but plausible, scenarios for black hole formation and growth with empirical correlations between luminosity and radio emission observed in low-redshift active galactic nuclei, we find that a model of black holes forming in molecular cooling halos is able to produce a 21 cm background that exceeds the cosmic microwave background (CMB) at z ≈ 17, though models involving larger halo masses are not entirely excluded. Such a background could explain the surprisingly large amplitude of the 21 cm absorption feature recently reported by the EDGES collaboration. Such black holes would also produce significant X-ray emission and contribute to the 0.5–2 keV soft X-ray background at the level of ≈10−13–10−12 erg s−1 cm−2 deg−2, consistent with existing constraints. In order to avoid heating the intergalactic medium (IGM) over the EDGES trough, these black holes would need to be obscured by hydrogen column depths of N H ~ 5 × 1023 cm−2. Such black holes would avoid violating constraints on the CMB optical depth from Planck if their UV photon escape fractions were below f esc 0.1, which would be a natural result of N H ~ 5 × 1023 cm−2 being imposed by an unheated IGM.