You should be able to book on any Avis website to get BA miles. There's normally the option to add your FF number on the final page of the booking process. If not, just show your card at the desk and get them to add it to your reservation.
As far as the BA AWD number goes, you (may) only have to use that if you want to get bonus miles/free something as part of a special promotion or if you want to add the free second driver, but you don't need it to get your standard booking miles. Whether Avis being able to identify that you're a Gold or Silver will lead to other benefits, who knows!
If there is a special BA + Avis promo you want to use, read the terms & conditions carefully to see what Avis sites it applies to and whether you need the AWD number.