A number of new tetraquark candidate states containing from one to four charm or anticharm quarks have been observed recently. Many of these new states have been discovered at the LHC. The production of these states via intrinsic charm in the proton is investigated. The tetraquark masses obtained in this approach, while dependent on the internal transverse momenta of the partons in the state, are shown to agree well with the measured masses. These calculations can provide some insight into the nature of the tetraquark candidates, whether as a bound meson pair or as a looser configuration of four individual partons. The kinematic distributions of these states as a function of rapidity and transverse momentum are also studied. The possible cross sections for these states are finally considered, with a comparison to the X(3872) pT distributions from p+p collisions at s=13 TeV. Published by the American Physical Society 2024