We report 18 dust continuum detections (≥3.3σ) at ∼88 and 158 $\mu{\rm m}$ out of 49 ultraviolet (UV)-bright galaxies (MUV < -21.3 mag) at $z$ > 6.5, observed by the Cycle-7 Atacama Large Millimeter/submillimeter Array (ALMA) Large Program, Reionization-Era Bright Emission Line Survey (REBELS) and its pilot programs. This has more than tripled the number of dust continuum detections known at $z$ > 6.5. Out of these 18 detections, 12 are reported for the first time as part of REBELS. In addition, 15 of the dust continuum detected galaxies also show a [C ii]$_{\rm 158\,{\rm \mu m}}$ emission line, providing us with accurate redshifts. We anticipate more line emission detections from six targets (including three continuum detected targets) where observations are still ongoing. We estimate that all of the sources have an infrared (IR) luminosity (LIR) in a range of $3\!-\!8 \times 10^{11}\, {\rm L_\odot }$, except for one with $L_{\rm IR} = 1.5^{+0.8}_{-0.5} \times 10^{12}\, \, {\rm L_{\odot }}$. Their fraction of obscured star formation is significant at ${\gtrsim} 50{{\ \rm per\ cent}}$, despite being UV-selected galaxies. Some of the dust continuum detected galaxies show spatial offsets (∼0.5-1.5 arcsec) between the rest-UV and far-IR emission peaks. These separations could imply spatially decoupled phases of obscured and unobscured star formation, but a higher spatial resolution observation is required to confirm this. REBELS offers the best available statistical constraints on obscured star formation in UV-luminous galaxies at $z$ > 6.5.