Ehhh, sure, it's the first date--but they've known each other for a while. And yes, they're inexperienced, but they're adults and independent.
For their case would there be a real point to delay beyond respecting some kind of convention?
For that matter, we don't actually know they're gonna. We assume it because real people at this point with their emotions and desires almost certainly would . . . but this is a manga and they might yet pull back.
Everyone is different on when they feel ready to have sex for the first time in a relationship, so I won't speak to what "normal" is.
However I think it goes against her established personality. She just realized she likes him a few hours ago. I don't remember if we have been told her age but she has said that she is mid aged which puts her at the very least in her early 30's but most people don't use that term until they're 40.
She has lived all that time with no interest in romance and seems to take everything very seriously.
So that person very nervously goes on her first date, falls, in love and has sex for the first time with her direct subordinate from work.
It doesn't really make a lot of sense.