How is Eowyn, shield maiden of Rohan written in Rohirric (is that the language of Rohan?)?
Edit - this answer is wrong
I've located a note in HoME12 stating that the East Gates of Moria "had borne Runic inscriptions in several tongues ... commands that all should depart who had not the leave of the Lord of Moria written in Quenya, Sindarin, the Common Speech, the languages of Rohan and of Dale and Dunland".
This makes it clear that (1) Rohirric was written, and (2) these inscriptions were obviously meant to be read by passers-by, so the Rohirrim in general could read.
Therefore I need to withdraw this answer.
I'm leaving it in it's original state below nonetheless as it may be useful to indicate that it's an incorrect interpretation of Aragorn's words in TT. But no more upvotes, please, and I'll gladly accept any downvotes anyone feels it necessary to give.
It's most likely not.
Rohan is very probably a pre-literate society, per Aragorn's remarks in the Two Towers:
There are only two references to any kind of writing in connection with the Rohirrim that I can find in LotR; the first one describing the floor of the Golden Hall:
And the second describing the horn that was given to Merry:
In the first case, it need not be so that the "runes" were actual writing; it could be just abstract symbols (which would be consistent with "...and strange devices", and similar to that described for the Druedain in Unfinished Tales: "They came no nearer to writing by their own invention than the use of a number of signs, for the most part simple, for the marking of trails or the giving of information and warning") or it could be a different language.
In the second case the horn was explicitly described as having been made by Dwarves in the following paragraph.
Rohirric could probably be adapted to a writing system (such as the Feanorian letters), and no doubt some of the more learned of the Rohirrim (such as Theoden, who spent time in Gondor during his youth) can read and write other languages, but Rohan would have no writing system of it's own.