As I don't have Polish citizenship, is it possible for me to buy a car under my name?
You can buy a car no problem but you'll need somewhere to register it. However, as a foreigner you can only register your car for as long as you are legally in Poland. In your case that will mean that you first have to get a residency permit (which will take at least 30 days) and then you can register your car, but only for a year or two (depending on how long a residency permit you were given). When your residency expires, so does your vehicle registration, meaning you need to renew both. In short, it's going to be far easier if your husband buys the car in his own name.