Are certain cards only available in certain missions, on certain difficulty levels? I only have 11 of the 24 permanent cards. I have 7/9 ships and have unlocked mission 11, but don't have all 4 medals for the first difficulty yet. I have all the 12 medals for all difficulty levels for the first 3 missions. On missions between 4-10 I'm on varying difficulties.
A couple tips to help with the card collection:
Obviously, fully upgrading your ship is very helpful to help with your survivability in general (needed to actually collect those cards). The Star Doubler IAP is pretty important to do this in a sensible amount of time (I wish I'd bought this sooner). And once you've finished with the upgrades, you can spend the left over stars buying a couple shields before mission start, to further increase your ability to survive. It's not a quick fix, but it does mean the same level gets easier over time.
Learn some levels that you can beat consistently on Insane difficulty and grind that for cards, ship parts and stars. For me this was Level 3 to begin with (as soon as I learned the "loop around the back" trick to dodge bullets, I could beat the boss fight every time). Later I switched to Level 7, because it doesn't have a boss fight at all, thought this level still requires some power up usage to be 100% consistent for me, so wait until you have the three starting shield+laser+nuke permanent cards. Remember that when you activate a shield, you can ram enemies (use this to clear those pesky choppers).
Note that you can only ever have one temporal card active at a time. So you can increase your odds of getting a permanent card, by playing while a temporal card is active. So even if the temporal card doesn't have a very useful effect, it's still 15 minutes of increased chance to get a permanent card.
Once you're very comfortable with a level and can beat it consistently on Insane with no damage, you can start using the Ace of Spades plane that increases card drop rates. Until then I'd suggest using the Iron Clad (for survivability) or Limp Berserker (the max fire rate helps keep the sky clear of enemies, so you are less likely to get swarmed - the low health requires knowing the level well though).
Finally, it seems that there's a special trick to getting the very last permanent card. I've seen people claim that you can only get it on level 8 Insane, while others claim it's actually whichever level is currently the last one (so 12 right now) on Insane. It's the only card I'm missing, so I'll update my answer once I've figured it out or gotten confirmation one way or another :)
Update: I managed to pick up (though not complete the level with) the last permanent on stage 11 insane, and managed to finally get it on stage 10 insane.
You need to play more to get all of the cards. I'm playing it since its release and still have one missing card. Skyforce Reloaded is designed to be played much more longer than Skyforce 2014. It will take much more time to collect every card and there will be new levels every once in a while, so completing all of the achievements will take longer, too.