Rapidly rotating classical OBe stars have been proposed as the products of binary interactions, and the fraction of Be stars with compact companions implies that at least some are. However, to constrain the interaction physics spinning up the OBe stars, a large sample of homogeneously analysed OBe stars with well-determined binary characteristics and orbital parameters are required. We investigate the multiplicity properties of a sample of 18 Oe, 62 Be, and two Of?p stars observed within the BLOeM survey in the Small Magellanic Cloud. We analyse the first nine epochs of spectroscopic observations obtained over approximately three months in 2023. Radial velocities (RVs) of all stars are measured. Applying commonly-used binarity criteria we classify objects as binaries, binary candidates, and apparently single (RV stable) objects. We further inspect the spectra for double-lined spectroscopic binaries and cross-match with catalogues of X-ray sources and photometric binaries. We classify 14 OBe stars as binaries, and an additional 11 as binary candidates. The two Of?p stars are apparently single. Two more objects are most likely currently interacting binaries. Without those, the observed binary fraction for the OBe sample (78 stars) is f_OBe_obs=0.18+/-0.04 (f_obs_cand=0.32+/-0.05 including candidates). This fraction is less than half of that measured for OB stars in BLOeM. Combined with the lower fraction of SB2s, this suggests that OBe stars have indeed fundamentally different binary properties than OB stars. We find no evidence for OBe binaries with massive compact companions, in contrast to expectations from binary population synthesis. Our results support the binary scenario as an important formation channel for OBe stars, as post-interaction binaries may have been disrupted or the stripped companions of OBe stars are harder to detect.