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Ultralight bosons, which are predicted in a variety of beyond-Standard-Model scenarios as dark-matter candidates, can trigger the superradiant instability around spinning black holes. This instability gives rise to oscillating boson condensates which then dissipate through the emission of nearly monochromatic gravitational waves. Such systems are promising sources for current and future gravitational-wave detectors. In this work, we consider minimally-coupled, massive vector bosons, which can produce a significantly stronger gravitational-wave signal compared to the scalar case. We adopt recently obtained numerical results for the gravitational-wave flux, and astrophysical models of black hole populations that include both isolated black holes and binary merger remnants, to compute and study in detail the stochastic gravitational-wave background emitted by these sources. Using a Bayesian framework, we search for such a background signal emitted using data from the first and second observing runs of Advanced LIGO. We find no evidence for such a signal. Therefore, the results allow us to constrain minimally coupled vector fields with masses in the range $0.8\times10^{-13}\mathrm{eV}\leq m_b\leq 6.0\times10^{-13}\mathrm{eV}$ at 95% credibility, assuming optimistically that the dimensionless spin distribution for the isolated black hole population is uniform in the range $[0,1]$. With more pessimistic assumptions, a narrower range around $m_b\approx 10^{-13}\mathrm{eV}$ can still be excluded as long as the upper end of the uniform distribution for dimensionless black hole spin is $\gtrsim 0.2$.