Based on 22.7 fb-1 of e+e- annihilation data collected at 33 different center-of-mass energies between 3.7730 and 4.7008 GeV with the BESIII detector at the BEPCII collider, Born cross sections of the two processes e+e-→φK+K- and e+e-→φKS0KS0 are measured for the first time. No indication of resonant production through an intermediate vector state V is observed, and the upper limits on the product of the electronic width Γe+e- and the branching fraction Br(V→φKK̄) of the processes e+e-→V→φK+K- and e+e-→V→φKS0KS0 at the 90% confidence level are obtained for a large parameter space in resonance masses and widths. For the current world average mass and width of the ψ(4230) of m=4.2187 GeV/c2 and Γ=44 MeV, we set upper limits on the φK+K- and φKS0KS0 final states of 1.75 and 0.47 eV at the 90% confidence level, respectively.