If you are in a cold climate, carpet is a great option. The "master bedroom and daughter's bedroom" tells me carpet would be just fine.

Remember: carpet hides HORRIBLE subfloors (can I assume this is a concrete slab?). And to get ANYTHING ELSE in there you need to do HUGE amounts of preparation. Vinyl is SUPER PICKY. If this is concrete, you will need to have the skills to handle concrete sanding/patching/moisture testing, etc.

In the end, vinyl can "work" if you have the skills to get the subfloor "smooth and flat as an ice-rink". If you are in a cold climate, the COLD of the basement will pull through VERY QUICKLY.

I suggest removing the carpet and assessing what needs to be done for the subfloor. And while you are down there in a basement without carpet...walk around in socks for an hour or so. How warm are your feet? That's about as good as vinyl gets (OK...it isn't THAT bad...but it is MUCH colder than carpet....much much much colder). And vinyl is LOUD. It allows noise to bounce and carry. You will find out how much the carpet was "doing" for you once it is gone. Go ahead and "live" with the bare floors for a week or two. You will see what I mean.

Photos of the bare subfloors (minus the carpet) would be very helpful for us.

Oh...BTW: carpet can be the cheapest floor to install - even with midgrade carpet and pad. Yes it requires installers....but it will still BEAT a vinyl floor purchase. And without the subfloor work.