Recently, a fully covariant version of the theory of F(T) torsion gravity has been introduced by M. Kršśák and E. Saridakis Classical Quantum Gravity 33, 115009 (2016). In covariant F(T) gravity, the Schwarzschild solution is not a vacuum solution for F(T)≠T, and therefore determining the spherically symmetric vacuum is an important open problem. Within the covariant framework, we perturbatively solve the spherically symmetric vacuum gravitational equations around the Schwarzschild solution for the scenario with F(T)=T+(α/2)T2, representing the dominant terms in theories governed by Lagrangians analytic in the torsion scalar. From this, we compute the perihelion shift correction to solar system planetary orbits as well as perturbative gravitational effects near neutron stars. This allows us to set an upper bound on the magnitude of the coupling constant, α, which governs deviations from general relativity. We find the bound on this nonlinear torsion coupling constant by specifically considering the uncertainty in the perihelion shift of Mercury. We also analyze a bound from a similar comparison with the periastron orbit of the binary pulsar PSR J0045-7319 as an independent check for consistency. Setting bounds on the dominant nonlinear coupling is important in determining if other effects in the Solar System or greater universe could be attributable to nonlinear torsion.