Does Mercy have polio? No!
We aren't told.
She could have. Polio existed at that time, but wasn't yet identified as such, so even if she had it, they wouldn't have identified it as such. (The book is set before the disease was well known as what it is.)
She also could have had other illnesses.
It dosnt really say in the book that she had polio but she could of so i personally think that she did but i have no prove nor does the book sorry if this didnt really answer your question.