We present broadband (gamma-ray, X-ray, near-infrared, optical, and radio) observations of the Swift gamma-ray burst (GRB) 090709A and its afterglow in an effort to ascertain the origin of this high-energy transient. Previous analyses suggested that GRB 090709A exhibited quasi-periodic oscillations with a period of 8.06 s, a trait unknown in long-duration GRBs but typical of flares from soft gamma-ray repeaters. When properly accounting for the underlying shape of the power-density spectrum of GRB 090709A, we find no conclusive (>3σ) evidence for the reported periodicity. In conjunction with the location of the transient (far from the Galactic plane and absent any nearby host galaxy in the local universe) and the evidence for extinction in excess of the Galactic value, we consider a magnetar origin relatively unlikely. A long-duration GRB, however, can account for the majority of the observed properties of this source. GRB 090709A is distinguished from other long-duration GRBs primarily by the large amount of obscuration from its host galaxy (A K,obs >~ 2 mag).