Is it consistent (from suitable large cardinals) that there is a ccc poset which forces PFA?
This seems quite implausible to me. If we could force PFA via ccc forcing, the ground model would have to be quite close to a model of PFA (having the correct continuum, no squares, SCH holding etc.). However, the ground model cannot be a model of full PFA (or even BPFA), since it follows from a result of Caicedo and Veličković that any ccc forcing over such models destroys BPFA.
The reference for the Caicedo-Veličković result is
Andrés Eduardo Caicedo and Boban Veličković, The bounded proper forcing axiom and well orderings of the reals, Math. Res. Lett. 13 (2006), no. 3, 393--408. (link)
They show that if $V$ and an inner model $M$ agree on $\omega_2$ and both satisfy BPFA then they have the same subsets of $\omega_1$. The conclusion above follows since any ccc forcing adds a subset of $\omega_1$.