(EDIT: until 30th September 2016, you can get a £30 sign-up credit when you join Zipcar if you use this link here.)
It wasn’t clear at the time what would happen if you were an existing Zipcar member. There is no option on the Zipcar site to add your BA details.
With my own Zipcar membership up for renewal this month, I emailed the company at firstname.lastname@example.org to ask.
It appears that they will not allow you to collect Avios points during your current membership year. However, as long as your provide the company with a BA or avios.com membership number before your renewal date, you will start collecting from that point.
I have now sent over my BAEC details and look forward to receiving a few miles from October onwards.