Can a non-inner automorphism map every subgroup to its conjugate?

Yes, the dihedral group of order 10 has this property. Its subgroup structure is very simple: $D_{10}$, ${e}$, the rotation subgroup, and the 5 subgroups generated by a flip. Any automorphism fixes the first three, so those are done, and shuffles the last five, and those 5 subgroups are all conjugate to each other.

All we need now is to show that there is a non-inner automorphism, but this is easy; the inner automorphisms always send a rotation to its inverse (or fix it) so we only need an automorphism which doesn't do that. Let the generators be $sigma,tau$, rotation and flip, and consider the automorphism defined on the rotations by $sigma mapsto sigma^2$ and fixing $tau$.