A Cat, a bat, a rat.

[dukealien]

All from the same family,
siblings by blood; the cat, the bat, the rat  colon (or em dash) and period here?  See below.
The cat sleeps on their chairs,
eats the food those people share could do without "the" and "those?"
With a nasty ear and a threatening snare,  for some reason I see "stare" here instead of "snare."
she wouldn't make it past May in this cold air perhaps "last" in place of "make it?"

I suppose the bat would sleep on chairs too  perhaps comma after "chairs"
but certainly wouldn't eat those people's food  em dash to connect with next line?
Too scared
They come around, she flits away, agenda and all  
right over to the next side of the people's wall  I picture a Mayan hut here, with one wall separating public porch and private after-court.

The rat, a blood-sucking, badly-behaved rabid animal  perhaps lose "a" here?
terrorized the poor cat
The rat originates from the other side of the wall
But she has no true home  is "true" necessary?
With light eyes and an expired relationship with the bat,  intriguing line
The rat roams about lose "about" here?
in search of home, not just a house
She has no purpose,
no one will love her.  but you just said... (great final line, though)

-VikaJigulina

Quite interesting.  You establish a slightly different grammar here, using capitalization that implies but does not show a period just before it.  You might try conventional, with periods (and occasional other marks of punctuation like colon and em dash), but the only real disadvantage is that the missing periods don't tell a reader when a longer pause is expected.  Keeps the reader on his toes, you do.

Most of the above suggestions (that's all they are, mind) are about removing words to improve or smooth the flow.  You may not want that - there's a patter-rhythm which can certainly be intentional.

(At the end, I only mean that you just stated right out that the rat's purpose is to find a home - then also state she has no purpose.  Perhaps you could bring down "agenda" from the previous stanza?)

On the whole, quite a nice concept.  I was a little confused by L5's "nasty" and "snare," but they're thought-provokingly different from the terms usually applied to cats.