"Ella está un poco cansada y quiere dormir."
Translation:She is a little tired and wants to sleep.
Yes, but that would make the word "sleep" a noun instead of a verb. In the sentence it's a verb, so you have to use "to" to make the English word also a verb.
To get sleep is a little different, it's like the difference between taking a showet and having a shower, but putting the word "get" might make it rather demanding and you would also have to change the spanish sentence.
I put : "She is a little tired and wants sleep", and it didn't accept it. Am I really wrong?
I'm pretty sure the use of "little" in this sentence is not good english. I don't know if there's a rule to that effect, but I've never seen it used that way. You would use "little" in front of a noun, e.g., "she is a little devil", "she is a little girl" and "bit" in front of an adjective: "she is a bit tired", "she is a bit angry". Or even, "she is a little bit tired".
Must be a regional thing, I say "little" in front of adjectives all the time.