Wolf-Rayet stars (WRs) are thought to be the final evolutionary stages of massive stars and immediate progenitors of stellar-mass black holes. Their multiplicity forms an important anchor point in single and binary population models for predicting gravitational-wave progenitors. Recent spectroscopic campaigns have suggested incompatible multiplicity fractions and period distributions for N- and C-rich Galactic WRs (WNs and WCs) at short as well as long orbital periods, in contradiction with evolutionary model predictions. In this work, we employed infrared interferometry using the $K$-band instrument GRAVITY at the VLTI to investigate the multiplicity of WRs at long periods and explore the nature of their companions. We present a survey of 39 Galactic WRs, including 11 WN, 15 WC and 13 H-rich WN (WNh) stars. We detected wide companions with GRAVITY for only four stars: WR 48, WR 89, WR 93 and WR 115. Combining with spectroscopic studies, we arrived at multiplicity fractions of $f^{\rm WN}_{\rm obs} = 0.55\pm0.15$, $f^{\rm WC}_{\rm obs} = 0.40\pm0.13$ and $f^{\rm WNh}_{\rm obs} = 0.23\pm0.12$. In addition, we also found other features in the GRAVITY dataset such as (i) a diffuse extended component in over half the WR sample; (ii) five known spectroscopic binaries resolved in differential phase data and (iii) spatially resolved winds in four stars: WR 16, WR 31a, WR 78 and WR 110. Our survey reveals a lack of intermediate (few 100s d) and long- (few years to decades) period WR systems. The 200-d peak in the period distributions of WR+OB and BH+OB binaries predicted by Case B mass-transfer binary evolution models is not seen in our data. The rich companionship of their O-type progenitors in this separation range suggest that the WR progenitor stars expand and interact with their companions, most likely through unstable mass-transfer, resulting in either a short-period system or a merger.