My clan has been through several wars and I can't figure out the systematic way that you unlock more "clan cards" in the war.
You earn cards after every collection day battle, and the number you get depends on your current league (Bronze, Silver, Gold, Legendary) along with your arena and whether you won a battle. For example, a person in Arena 11, in a Bronze II clan will get 140 cards for every Collection Day battle won and only 70 cards for every Collection Day battle lost. Consequently, if you want your cards to reach tournament standard (the usual card levels for Bronze league), you're going to need as much clan participation as possible. A good strategy to maximize your cards earned is to pick the kinds of events that you specifically are good at (i.e. if you're good at Sudden Death, choose it whenever possible), and to avoid ones that you aren't good at.
Your clan card levels are also capped at whatever level your cards are at, and if you don't have one of the clan cards, you can't use it in your war deck.
The only way to get cards is to battle. Make sure everyone in your clan has used up all three of their attacks, and won as many as possible. You and your clanmates being in a higher arena will also help as you get more cards per battle and the more people you have, the more attacks you'll get so the more cards you'll obtain.
They are the only ways I can offer of getting more cards. My clan didn't seem to get many cards at first, but we got lots more once we made it open, instead of invite only.