Uly, can we use 'chaperon' here rather then escort?

I think so.

https://books.google.com/ngrams/graph?content=escort+%2A+to+the+hospital%2Cchaperone+%2A+to+the+hospital&year_start=1800&year_end=2019&corpus=26&smoothing=3&direct_url=t2%3B%2Cescort%20%2A%20to%20the%20hospital%3B%2Cc0%3B%2Cs0%3B%3Bescort%20her%20to%20the%20hospital%3B%2Cc0%3B%3Bescort%20you%20to%20the%20hospital%3B%2Cc0%3B%3Bescort%20him%20to%20the%20hospital%3B%2Cc0%3B%3Bescort%20them%20to%20the%20hospital%3B%2Cc0%3B%3Bescort%20me%20to%20the%20hospital%3B%2Cc0%3B%3Bescort%20us%20to%20the%20hospital%3B%2Cc0

Though you said "I think so" the ngram doesn't show any use of it. Would it be natural/correct to say - I want you to chaperone her to the hospital
?

The Ngram just confirmed my doubts))

Well, it has a lot of examples of ''chaperone * to the...''
but none of them is a hospital. Looks like we can chaperone people to many different places but the hospital?

yes, chaperones normally accompany young women in the company of young men to make sure there’s no hanky-panky.

ah, I see.
I like hanky-panky expression ))) Sounds funny.