I absolutely love gathering quotes and trying to come to a solid conclusion about something that SRD never flat out said one way or the other. Even if that conclusion is solid only in <I>my</I> mind.

A psychologist might have something to say about my need to have everything answered so clearly, but... Anyway, here's my conclusions on this topic.
The <I>Haruchai</I> need sleep.
Aside from Bannor needing sleep <I>after</I> the Vow was abandoned, in LFB, he said:
"No Bloodguard has slept since the <I>Haruchai</I> swore their Vow."
There's no way to interpret that other than that the <I>Haruchai</I> sleep. It seems to me that the <I>Haruchai</I> do everything they do out of the extremity of their emotions, or out of need. So they sleep because either it is a deeply meaningful act to them, or they need to sleep. I'm going with need.
Navarino, you may be right that they don't sleep in the 2nd Chronicles. (Or at least Cail doesn't sleep after Brinn becomes <I>ak-Haru Kenaustin Ardenol</I>. It's always possible that they took turns sleeping when Ceer and Hergrom were alive. I imagine they could wake each other easily enough with their mind-speech.) The specific passage that I assume you're referring to is:
Mistweave lay felled in his canvas cover as if during the night his desire to match Cail had suffered a defeat. Only Cail, the Demondim-spawn, and Findail were on their feet.
Technically, we don't <I>know</I> that Cail didn't sleep. It's conceivable that he slept, but for less time than Mistweave, and woke up first. And we don't know about other nights either. But since Cail is grouped with Vain and Findail (I don't imagine anyone suspects they sleep) in contrast to Mistweave, I think SRD wants us to think that Cail doesn't sleep.
So the question is, why doesn't Cail need to sleep? Has there been a change in the <I>Haruchai</I> since the 1st Chronicles? I doubt it. I think such a fundamental change would have been pointed out. Covenant would have been made aware of it - maybe Brinn would have said, "The Vow removed the need for sleep from the Bloodguard. Now, no <I>Haruchai</I> need sleep" - and we would have heard him talking or thinking about the difference/implications/horror of the situation.
It's possible that they were in a kind of mini-Vow situation, and the Earthpower took away their need for sleep again, as it did for the Bloodguard. But we probably would have witnessed something like "<I>the rocks on which they stood thundered, and fire ran through them, sealing their bones to the promise they had made</I>" if that was the case. And Covenant certainly would have I given us a "Hellfire!" if that had happened.
I think Cail just decided to not sleep. As I said last time, being such extraordinary physical specemins, I'm sure the <I>Haruchai</I> don't need nearly the sleep we do. And that's even if they didn't have a strong connection to the Earthpower. (Which, as Damelon reminds us, they do. I love it when Mhoram said, <I>"In their way, they know the name of the Earthpower more surely than any Lord."</I>) As I see it, the absolute nature of their character, the way they commit completely to a course of action (I love phrases like "out of the extremity and innocence of their hearts" and "extravagant <I>Haruchai</I>"), allows them to put off sleep for loooooong periods of time. It must become more and more difficult as time goes by, but we would, of course, never see any sign of their effort.
Damelon wrote:ak-Haru Kenaustin Ardenol always seemed to me to be a creation of a different kind of vow than the one that the Bloodguard made.
My thinking is that this guy named <I>Kenaustin Ardenol</I> became <I>ak-Haru</I> by attaining perfection. The <I>Haruchai</I> definition of perfection, that is. I never thought that he took any vow to become the <I>Haruchai</I> legend that he is, just that he is the ultimate example of all they desire. And it was when he achieved perfection that he gained the abilities, presumably granted to him by the Earthpower, that the other <I>Haruchai</I> do not have. Although you could be right. It could be a vow of some sort. This thread <I>is</I> called "Speculation" after all.

[I guess he made a vow to Berek though. But I think he was <I>ak-Haru</I> before he met Berek. More speculation.]
Damelon wrote:It seems that when the <I>Haruchai</I> make an oath, the Earthpower takes notice.
Whether or not <I>ak-Haru Kenaustin Ardenol</I> took any vow, that's a big AMEN!
Sooooooooo, there's my latest overthinking of, and rambling on about, the <I>Haruchai</I>.

I can live with these answers.