  In the framework of $f(T)$ theories of gravity, we solve the field equations
for $f(T)=T+\alpha T^{n}$, in the weak-field approximation and for spherical
symmetry spacetime. Since $f(T)=T$ corresponds to Teleparallel Gravity, which
is equivalent to General Relativity, the non linearity of the Lagrangian are
expected to produce perturbations of the general relativistic solutions,
parameterized by $\alpha$. Hence, we use the $f(T)$ solutions to model the
gravitational field of the Sun, and exploit data from accurate tracking of
spacecrafts orbiting Mercury and Saturn to infer preliminary insights on what
could be obtained about the model parameter $\alpha$ and the cosmological
constant $\Lambda$. It turns out that improvements of about one-three orders
with respect to the present-day constraints in the literature of magnitude seem
possible.
