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Using a data sample of 427.9 fb\\(^-1\\) collected by the Belle~II detector at or near the \\((4S)\\) and \\((10753)\\) resonances, the cross sections for \\(e^+e^- h^+h^-J/\\) \\((h=/K/p)\\) at center-of-mass energies ranging from 3.8 GeV or the production threshold to 5.5/6.0/7.0 GeV have been measured via initial-state radiation. The cross sections for the processes \\(e^+e^- ^+^-J/\\) and \\(e^+e^- K^+K^-J/\\) are consistent with previously published results. The cross sections for these channels obtained by combining with previous Belle results are also given. The process \\(e^+e^- p p J/\\) is investigated for the first time. The yields are small and no significant structure is observed in the cross section versus energy. Searches for vector charmonium-like states in the \\(h^+h^-J/\\) systems, and for associated intermediate states in the \\(h^ J/\\) systems, are also presented.