Pretty much the question in the title: can there be a well-ordering the reals $\langle r_\alpha\mid \alpha<\kappa \rangle$, all of whose initial segments $I_\beta := \{r_\alpha\mid \alpha<\beta<\kappa\}$ are Lebesgue-measurable?
Observation: if the Continuum Hypothesis holds, then any well-ordering of the reals in ordertype $\omega_1$ will have this property. So it's consistent that such a well-ordering exists.
Partial answer: if we additionally require that the order-type of the well-ordering is a cardinal, then this is independent of ZFC. (Edit: see comment below by Farmer S on how to turn this into a full answer)
For example, add $\aleph_2$ random reals to $L$. In the extension, $\mathbb{R}^L$ is a set of size $\aleph_1$, with positive outer measure. Any well-ordering of the reals in type $\omega_2$ will have a proper initial segment containing all of $\mathbb{R}^L$. This means that the initial segment (if measurable) needs to have positive measure and hence cardinality $2^{\aleph_0}$. So no well-ordering can have all its initial segments being measurable in this model.
Another way to see that there can be no such well-ordering in this model is by noticing that the set of random reals added must be a Sierpinski set (every measurable subset is countable). Again, since any well-ordering in type $\omega_2$ must have an initial segment containing $\aleph_1$ many random reals, any such initial segment will have positive measure (if measurable) and hence cardinality $2^{\aleph_0}$.