I was going to do a little rhyme here, but turns out I can't. So the question is, does that disqualify me for zebrahood?

Nope - a stylised speech pattern is one of the marks of a shamanic order, and one they defend nearly as fiercely as their recipes. (Applying the pattern to a foreign language is one of Zecora's more minor heresies.)

38. Do all zebras rhyme?
No. Either it's a trait common to shaman or else its a personal geas of Zecora's.

7038839
Ooh I like this one! I was going to use it in a fic waaaay back when, but it doed, sadly. Had some good rhymes in it.

Honestly, I've always interpreted the rhyming as her accent.

Like... it's some bizarre magical thing wherein that's just what it sounds like when she speaks Equestrian.

But of course, the "is zebras real" question throws a wrench into this. It's possible they do, it depends on how much we can extrapolate zebra culture from a single example who may very well not even be part of it.

7038992
I think you are possibly the first person to ask that question.

7034519 Well apparently, the writers said no. Zecora was unique in that she fell through a portal and emerged from it speaking in rhyme.

7039029
I've now heard that a few times today - was this in a recent interview?

7039035 Yes, I believe it was with Jim Miller, actually. I think that would've been interesting to see.

As a society, yes they rhyme everything because once something like this gets started, it's really fun, and it takes a really pissy person to demand they stop, and since there isn't anyone to do that in Equestria, it has become their thing.

I like her rhyming, I'm just horrible at it myself. Head-canon-wise... I'm honestly not sure. (Read: 'undecided'.) It's a nice little quirk to add to Zebrian shamans in general. Might be a tradition. Might be something that's actually useful 'back home', somehow. But it might be, just as well, just a personal quirk of her as an individual. Keeping your mind on your toes, so to speak. Adding a flourish to your prose. Being more aware of your language, and how you use it. There can be power in words. Especially in a world with magic. So maybe, her constant rhyming is a way of training her mind. Her focus and concentration.
Maybe.