  We explore the role of galactic feedback on the low redshift Lyman-$\alpha$
(Ly$\alpha$)~forest ($z \lesssim 2$) statistics and its potential to alter the
thermal state of the intergalactic medium. Using the Cosmology and Astrophysics
with Machine Learning Simulations (CAMELS) suite, we explore variations of the
AGN and stellar feedback models in the IllustrisTNG and Simba sub-grid models.
We find that both AGN and stellar feedback in Simba play a role in setting the
Ly$\alpha$ forest column density distribution function (CDD) and the Doppler
width ($b$-value) distribution. The Simba AGN jet feedback mode is able to
efficiently transport energy out to the diffuse IGM causing changes in the
shape and normalization of the CDD and a broadening of the $b$-value
distribution. We find that stellar feedback plays a prominent role in
regulating supermassive black hole growth and feedback, highlighting the
importance of constraining stellar and AGN feedback simultaneously. In
IllustrisTNG, the AGN feedback variations explored in CAMELS do not affect the
Ly$\alpha$ forest, but varying the stellar feedback model does produce subtle
changes. Our results imply that the low-$z$ Ly$\alpha$ forest can be sensitive
to changes in the ultraviolet background (UVB), stellar and black hole
feedback, and that AGN jet feedback in particular can have a strong effect on
the thermal state of the IGM.
