Rotating neutron stars (NSs) are promising sources of gravitational waves (GWs) in the frequency band of ground-based detectors. They are expected to emit quasi-monochromatic, long-duration GW signals, called continuous waves (CWs), due to their deviations from spherical symmetry. The degree of such deformations, and hence the information about the internal structure of a NS, is encoded in a dimensionless parameter ๐ called ellipticity. Searches for CW signals from isolated Galactic NSs have shown to be sensitive to ellipticities as low as ๐ โผ O (10 โ9 ). These searches are optimal for detecting and characterising GWs from individual NSs, but they are not designed to measure the properties of NSs as population, such as the average ellipticity ๐ av . These ensemble properties can be determined by the measurement of the stochastic gravitational-wave background (SGWB) arising from the superposition of GW signals from individually-undetectable NSs. In this work, we perform a cross-correlation search for such a SGWB using the data from the first three observation runs of Advanced LIGO and Virgo. Finding no evidence for a SGWB signal, we set upper limits on the dimensionless energy density parameter ฮฉ gw ( ๐ ). Using these results, we also constrain the average ellipticity of Galactic NSs and five NS "hotspots", as a function of the number of NSs emitting GWs within the frequency band of the search ๐ band . We find ๐ av < โผ 1.8 ร 10 โ8 , with ๐ band = 1.6 ร 10 7 , for Galactic NSs, and ๐ av < โผ [3.5 โ 11.8] ร 10 โ7 , with ๐ band = 1.6 ร 10 10 , for NS hotspots.