"Please wait while sitting on a chair."
There is no while in this sentence. I would assume that other answers will be accepted, as they should be. 'Please have a seat and wait' or 'Please sit in the chair and wait' would be more accurate.
Also, it is probably better to sit 'in a chair' than 'on a chair'. Better for the back.
While is generally represented through the use of verb root + ながら, as in あるきながら 'while walking'.
Yeah... this sentence is very awkward in English. I can't point to a grammar rule or anything exactly, but I think it feels awkward to pair "wait" with another verb, but put "wait" first. That is, "Sit down and wait." sounds natural, but "Wait while you sit." is very strange. I think since the connotation for "wait" is stopping all other actions, in English at least it's best to frame it as: complete action 1, and then "wait".
But I guess for just teaching translation here on Duolingo the sentence is mostly ok. Except I agree with the other commenter that "sitting in a chair" should definitely be the default sentence (it's much more common phrasing and natural sounding), and NOT "sitting on a chair".**
The difference between what you wrote and what DuoLingo wanted is the emphasis on waiting.
The final verb used in the sentence is assumed to be the priority verb. In English, for example, we would not say 'Please wait and sit', unless the emphasis was on the sitting.
Your answer emphasized the request to sit down as opposed to waiting. In Japanese, the main request can be encapsulated as 「待ってください」('Please wait'). Expanded, this is 「座って待ってください」, or 'Please sit and wait.'.
Due to your answer containing the verb at the beginning of the sentence, paraphrased, is more akin to 'Hold on, please sit down.'
Lastly, 待っていす would be ungrammatical. A verb cannot be placed conjugated to the て form directly before a noun. If you wer to insert a comma (待って、いすにすわってください）, it would be fine grammatically but still have a meaning different from the English presentation.