Single-phase concentrated solid-solution alloys (CSA), i.e., alloys without a principle alloying element but one randomly populated crystal structure, exhibit attractive material properties such as very high ductility at cryogenic temperatures, a gentle decrease of strength with temperature, or an unexpectedly high resistance against irradiation. For clarification of those observations assessment of atomic transport mechanisms including formation and migration of equilibrium point defects is indispensable. Positron annihilation lifetime spectroscopy measurements are performed to quantify the concentration of quenched-in thermal vacancies in fcc CSAs after quenching from temperatures close to their onset of melting. For various alloy compositions the concentration of quenched-in vacancies decreases with increasing entropy of mixing ΔSmix. Whereas alloys with three constituents in nonequimolar fractions (CrFeNi) exhibit vacancy concentrations in the 10-5 range, the studied alloys with four (CoCrFeNi) and five constituents (CoCrFeMnNi, AlCoCrFeNi) do not show a vacancy-specific positron lifetime. Therefore, the concentration of quenched-in vacancies must be in the range of 10-6 or less. It can be concluded that there is either only a vanishingly small fraction of vacancies present at temperatures near the onset of melting or the generated vacancies are inherently unstable.