Where or how can I process the permit residence for Poland?
You apply at the foreigners office for your Voivodeship or at branch offices thereof. For example, if you live in Mazowiecki, you can apply at the main office, i.e. ul. Dluga 5 in Warsaw. Alternatively you can submit your paperwork at certain 'branch offices' or by post. There's a list of such places here: mazowieckie.pl/en/for-foreigners/residence/28,Residence-legalization.html
However, by far and away your best bet is to go to the main office (that way there is at least a chance that they will tell you what is wrong with your application so you can correct it there and then rather than having it posted back to you with a request that you correct it).
However, the vehicle registration office don't actually want to see your residency permit; they want to see your zameldowania, your 'maldunek'. You get one of those from the gmina office for where you live by showing them your residency permit. Your maldunek will only be valid for as long as your residency permit is valid and your vehicle registration is only going to be valid for as long as your maldunek is valid.
As I said before, it is going to be far far easier for your husband to buy the car in his own name than have it registered in your name.