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A stochastic gravitational-wave background (SGWB) is expected to be produced by the superposition of individually undetectable, unresolved gravitational-wave (GW) signals from cosmological and astrophysical sources. Such a signal can be searched with dedicated techniques using the data acquired by a network of ground-based GW detectors. In this work, we consider the astrophysical SGWB resulting from pulsar glitches, which are sudden increases in the rotational pulsar frequency, within our Galaxy. More specifically, we assume glitches to be associated with quantized, superfluid, vortex-avalanches in the pulsars, and we model the SGWB from the superposition of GW bursts emitted during the glitching phase. We perform a cross-correlation search for this SGWB-like signal employing the data from the first three observation runs of Advanced LIGO and Virgo. Not having found any evidence for a SGWB signal, we set upper limits on the dimensionless energy density parameter $Ω_{\mathrm{gw}}(f)$ for two different power-law SGWBs, corresponding to two different glitch regimes. We obtain $Ω_{\mathrm{gw}}(f)\leq 7.5 \times 10^{-10}$ at 25 Hz for a spectral index 5/2, and $Ω_{\mathrm{gw}}(f)\leq 5.7 \times 10^{-17}$ at 25 Hz for a spectral index 17/2. We then use these results to set constraints on the average glitch duration and the average radial motion of the vortices during the glitches for the population of the glitching Galactic pulsars, as a function of the Galactic glitch rate.