It is not uncommon for them to not ask for a card, no worries there. I have never used that site, I have used carrentalsavers.com, but maybe for peace of mind you could run your reservation number through
Avis and see if it comes up? I always did that. I made a reservation and then cancelled it quite often according to price (I think I did it maybe 6 times before getting a price I was happy with) then would check the reservation on the site of the actual company (I ended up using Dollar but I do believe I booked Avis at one point). Just be sure to enter the proper number, the Avis number not the rentalcars number, when checking but you should be able to see all of the details of your reservation which should put your mind at ease.