Given a (graded) ring $R$, to define a formal group law it is equivalent to define a ring homomorphism $\phi:L\to R$ where $L$ is Lazard's ring. Is there any notion of defining a formal group law on ring spectrum $E$, i.e. a map $f:\mathbb{H}L\to E$? Would this in any way extend the notion of formal group laws on algebraic rings or the Landweber Exact Functor Theorem?

I think the right correspondence is the one mentioned in the comments. The main reason is that if you have a map of ring spectra $HL \to E$ then $E$ is an EilenbergMaclane spectrum. So in particular, you are only getting information about formal group laws over EM spectra and not any other kinds of spectra. However, it might be interesting to compute this. It is easy not easy to do or it is trivial. The main sources for computing this type of mapping space (maps of commutative ring spectra) that I know of is the obstruction theory of Goerss and Hopkins. There could be others though. 

