The speech in the OP is Théoden's speech from the movie. In the book, he says,

Arise, arise, Riders of Théoden! Fell deeds awake: fire and slaughter! spear shall be shaken, shield be splintered, a sword-day, a red day, ere the sun rises! Ride now, ride now! Ride to Gondor!

and just a page or two later

‘To me! To me!’ cried Théoden. ‘Up Eorlingas! Fear no darkness!’

(ROTK book 5, chapters 5 & 6)

Now, Eomer does say something very similar to Theoden's speech from the Charge of the Rohirrim, upon discovering his sister and assuming her dead:

‘Éowyn, Éowyn!’ he cried at last. ‘Éowyn, how come you here? What madness or devilry is this? Death, death, death! Death take us all!’

and in the next paragraph

‘Death! Ride, ride to ruin and the world’s ending!’

(later in ROTK book 5, chapter 6)

The line "Forth Eorlingas!" is Theoden's (though the Rohirrim also say it en masse), but it's from the chapter about Helm's Deep and the one preceding it. (TTT book 3, chapters 6 & 7).

And, look, Theoden's speech before the Charge of the Rohirrim at Pelennor is a cinematic masterpiece. And Bernard Hill (RIP) is probably the biggest reason why; his delivery is unimpeachable. In a lot of ways, the speech in the movie is better than what's in the books at that moment; but honestly they are trying to do different things.

So technically, the movie version is a bit heightened from the book version, and cobbled together from a few different sources. To put it all together:

Arise, arise, Riders of Théoden!

Verbatim from Theoden's speech in ROTK chapter 5. Correctly placed in the narrative.

Spear shall be shaken, shield shall be splintered,

Also from Theoden's speech in ROTK chapter 5, but not quite verbatim (they added a "shall" in there, presumably for flow...or maybe just Bernard Hill forgot the exact line). Correctly placed in the narrative, though they left out "Fell deeds awake: fire and slaughter!" before this line.

a sword-day, a red day, ere the sun rises!

Verbatim from Theoden's speech in ROTK chapter 5. Correctly placed in the narrative.

Ride now, ride now, ride!

Also from Theoden's speech in ROTK chapter 5, with an extra "Ride!" thrown in. Correctly placed in the narrative.

Ride for ruin and the world's ending! Death! Death! Death!

Transplanted from Eomer's speech in ROTK chapter 6, with "to" replaced with "for," the order of the lines reversed, and the mood changed from despair to defiance.

Forth Eorlingas!

Right character, wrong moment. Transplanted from Theoden's speech in TTT chapter 7 at Helm's Deep, but in the same vein as Theoden's shout of "Up Eorlingas!" from the beginning of ROTK chapter 6.

So Kelly, from the original tweet, probably overheard her husband reading Chapter 5 of ROTK, and her brain filled in the movie version of the quote while she was writing the tweet.