Laser-driven compact particle accelerators can provide ultrashort pulses of broadband X-rays, well suited for undertaking X-ray absorption spectroscopy measurements on a femtosecond timescale. Here the Extended X-ray Absorption Fine Structure (EXAFS) features of the K-edge of a copper sample have been observed over a 250 eV window in a single shot using a laser wakefield accelerator, providing information on both the electronic and ionic structure simultaneously. This capability will allow the investigation of ultrafast processes, and in particular, probing high-energy-density matter and physics far-from-equilibrium where the sample refresh rate is slow and shot number is limited. For example, states that replicate the tremendous pressures and temperatures of planetary bodies or the conditions inside nuclear fusion reactions. Using high-power lasers to pump these samples also has the advantage of being inherently synchronised to the laser-driven X-ray probe. A perspective on the additional strengths of a laboratory-based ultrafast X-ray absorption source is presented.