  We apply Gaussian processes (GP) in order to impose constraints on
teleparallel gravity and its $f(T)$ extensions. We use available $H(z)$
observations from (i) cosmic chronometers data (CC); (ii) Supernova Type Ia
(SN) data from the compressed Pantheon release together with the CANDELS and
CLASH Multi-Cycle Treasury programs; and (iii) baryonic acoustic oscillation
(BAO) datasets from the Sloan Digital Sky Survey. For the involved covariance
functions, we consider four widely used choices, namely the square exponential,
Cauchy, Mat\'{e}rn and rational quadratic kernels, which are consistent with
one another within 1$\sigma$ confidence levels. Specifically, we use the GP
approach to reconstruct a model-independent determination of the Hubble
constant $H_0$, for each of these kernels and dataset combinations. These
analyses are complemented with three recently announced literature values of
$H_0$, namely (i) Riess $H_0^{\rm R} = 74.22 \pm 1.82 \,{\rm km\, s}^{-1} {\rm
Mpc}^{-1}$; (ii) H0LiCOW Collaboration $H_0^{\rm HW} = 73.3^{+1.7}_{-1.8}
\,{\rm km\, s}^{-1} {\rm Mpc}^{-1}$; and (iii) Carnegie-Chicago Hubble Program
$H_0^{\rm TRGB} = 69.8 \pm 1.9 \,{\rm km\, s}^{-1} {\rm Mpc}^{-1}$.
Additionally, we investigate the transition redshift between the decelerating
and accelerating cosmological phases through the GP reconstructed deceleration
parameter. Furthermore, we reconstruct the model-independent evolution of the
dark energy equation of state, and finally reconstruct the allowed $f(T)$
functions. As a result, the $\Lambda$CDM model lies inside the allowed region
at 1$\sigma$ in all the examined kernels and datasets, however a negative slope
for $f(T)$ versus $T$ is slightly favored.
