I feel your pain - while I can't speak (or read for the matter) Korean I come from non ASCII (or rather greatly extended ASCII) language background.
Without delving (too much) into character encoding hell: using unicode escape characters seems to be the way to go. These can be generated with the help of stringi::stri_escape_unicode().
So if I were in your shoes I would consider something along these lines / note that since my understanding of Korean is very limited, and your function is somewhat language specific, I can't actually confirm or rule out my approach.
sampleprob <- function(url) {
# sample url: "http://dart.fss.or.kr/dsaf001/main.do?rcpNo=20200330003851"
# stringi::stri_escape_unicode("연결재무제표 주석")
result <- grepl("\\uc5f0\\uacb0\\uc7ac\\ubb34\\uc81c\\ud45c \\uc8fc\\uc11d", html_text(read_html(url)))
return(result)
}