You can fit a new front wheel w/o any issues.
Rear will pose some challenges. A cassette rear will be wider than the original. You’ll need to pry the rear stays apart to get the wheel in. Often doable but not ideal. Unless you want to cold-set - permanently spread the frame - and realign the dropouts, you’re probably better off looking for a freewheel rear to equip with the same number of gears as your current wheel.
The big question is, if you want to keep the classic look, why are you thinking about new wheels?
Nothing has changed that has made modern wheels noticeably less prone to flats.
Find out what’s causing your flats, fix the issue and ride on.
Do you have a good pump with a pressure gauge?
It can be as easily explained as riding underinflated.
Skinny bike tires are nothing like car tires WRT pressure retention. They’ll need topping up about weekly.
And you’d better overcome your aversion against flat fixing.
As long as you ride inflatable tires, flats are ALWAYS an option.