In the vanadium oxyfluoride compound (NH₄)2[C₇H₁₄N][V₇O₆F₁₈] (DQVOF), the V4⁺ (3d¹, S=1/2) ions realize a unique, highly frustrated breathing kagome lattice composed of alternately sized, corner-sharing equilateral triangles. Here we present an ¹⁷O NMR study of DQVOF, which isolates the local susceptibility of the breathing kagome network. By a fit to series expansion, we extract the ratio of the interactions within the breathing kagome plane, J_∇/J_Δ = 0.55(4), and the mean antiferromagnetic interaction J = 60(7) K. Spin lattice (T₁) measurements reveal an essentially gapless excitation spectrum with a maximum gap Δ/J = 0.007(7). Our study provides new impetus for further theoretical investigations in order to establish whether the gapless spin liquid behavior displayed by DQVOF is intrinsic to its breathing kagome lattice or whether it is due to perturbations to this model, such as a residual coupling of the V⁴⁺ ions in the breathing kagome planes to the interlayer V³⁺ (S=1) spins.