Characterizing the epoch of reionization (EoR) at $z\gtrsim 6$ via the redshifted 21 cm line of neutral Hydrogen (HI) is critical to modern astrophysics and cosmology, and thus a key science goal of many current and planned low-frequency radio telescopes. The primary challenge to detecting this signal is the overwhelmingly bright foreground emission at these frequencies, placing stringent requirements on the knowledge of the instruments and inaccuracies in analyses. Results from these experiments have largely been limited not by thermal sensitivity but by systematics, particularly caused by the inability to calibrate the instrument to high accuracy. The interferometric bispectrum phase is immune to antenna-based calibration and errors therein, and presents an independent alternative to detect the EoR HI fluctuations while largely avoiding calibration systematics. Here, we provide a demonstration of this technique on a subset of data from the Hydrogen Epoch of Reionization Array (HERA) to place approximate constraints on the IGM brightness temperature. From this limited data, at $z=7.7$ we infer "$1\sigma$" upper limits on the IGM brightness temperature to be $\le 316$ "pseudo" mK at $\kappa \parallel =0.33"pseudo"h$ Mpc$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ (data-limited) and $\le 1000$ "pseudo" mK at $\kappa \parallel =0.875"pseudo"h$ Mpc$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ (noise-limited). The "pseudo" units denote only an approximate and not an exact correspondence to the actual distance scales and brightness temperatures. By propagating models in parallel to the data analysis, we confirm that the dynamic range required to separate the cosmic HI signal from the foregrounds is similar to that in standard approaches, and the power spectrum of the bispectrum phase is still data-limited (at $\gtrsim 106$ dynamic range) indicating scope for further improvement in sensitivity as the array build-out continues.