We present detailed multifrequency, multiepoch radio observations of GRB 140304A at z = 5.283 from 1 to 86 GHz and from 0.45 to 89 days. The radio and millimeter data exhibit unusual multiple spectral components, which cannot be simply explained by standard forward and reverse shock scenarios. Through detailed multiwavelength analysis spanning radio to X-rays, we constrain the forward shock parameters to Ek,iso 4.9 × 1054 erg, A * 2.6 × 10−2, ϵ e 2.5 × 10−2, ϵ B 5.9 × 10−2, p 2.6, and θ jet 1 1, yielding a beaming-corrected γ-ray and kinetic energy, E γ 2.3 × 1049 erg and E K 9.5 × 1050 erg, respectively. We model the excess radio emission as due to a combination of a late-time reverse shock (RS) launched by a shell collision, which also produces a rebrightening in the X-rays at 0.26 days, and either a standard RS or diffractive interstellar scintillation (ISS). Under the standard RS interpretation, we invoke consistency arguments between the forward and reverse shocks to derive a deceleration time, tdec 100 s, the ejecta Lorentz factor, Γ(tdec) 300, and a low RS magnetization, RB 0.6. Our observations highlight both the power of radio observations in capturing RS emission and thus constraining the properties of GRB ejecta and central engines and the challenge presented by ISS in conclusively identifying RS emission in GRB radio afterglows.