Shotguns are nice, as mentioned, early on. For most anti-armour weapons, just look at the "pirate'y" line of weapons for all your early game needs. They're a little bit inaccurate, you'll definitely get better guns, but they're all amazing close ranged weapons. You might have a bit of difficulty finding ammo at the beginning, but as soon as you can make it, it's very fast and cheap to do so. They'll definitely get you through to mid-game, where your options open up dramatically (including other types of ammo for these guns, if you do decide to keep them around).
Pirate'y line is:
Assault Cannons: great early game long-distance grenade launcher. Normal cannon balls are OK for anti-armour with 80 damage and arcing shots, 60 damage HE balls are good for anything else, and are often good against medium armour as well. Pyro balls aren't bad either. Gets lots of types of ammo later on. The defining factor of whether you can take on harder missions early on is whether you've got strong gals, an assault cannon or two, and the HE ammo for them. Note: you can conga-line a single cannon with multiple gals if you have to. That initial cannon is a godsend. Once you realize that you're getting 1-4 kills/stuns every map with that gal, and your brain overcomes the "but I could only shoot once a turn" bit of them, you'll realize how good they are. HE rounds are waayyyy more accurate than their stats suggest.
Blunderbuss: it's a very big, very close-ranged shotgun. 6 pelletx25 damage normal shells alleviates lots of problems with light-med armour. Also gets 40 damage HE shells later on. A four barreled auto-shot can and will mulch things that shotguns probably shouldn't be able to, or strip tank shields like nothing else. If you could get into melee and take 2-4 swings, then you can probably single/auto fire a blunderbuss from 4-6 squares away as well, with very similar results.
Hand Cannon: it's not really that accurate, but it hits very hard. 42+(0.15bravery) gives you about 50 damage a shot, so even with bad armour piercing, the damage often overcomes it anyway. Also gets 52 damage HE rounds, which are quite fun. It's a pistol, and even with only four shots and low'ish accuracy, is the best non-training pistol you'll get for a long time.
Handle: still one of the best stun weapons in the game. Everyone should have one of these, or a cattle prod, for a very long time.
Everything melee'y: melee weapons are piratey. There's good early ones, there's bad early ones, but no matter what stats a gal has, there's a weapon she'll either be accurate with, hit hard with, or both. While -armour on weapons is nice, so are fast weapons, or hard hitting weapons. There's always something to use, so no cause to complain there.
If you're moderately well kitted, it's often worthwhile taking on a necropirates mission, just to get more of these things. They might be low on ammo and accuracy, but they'll punch through most things early on, and you get them pretty early (and even start with a few). There's even research chains to get better versions of them, or better ammo for them, but they might be outclassed by the time you do. Still, you can't ever say that you don't have anti-armour options in x-piratez, from day two'ish (there's like 1-3 research topics and you get tonnes for them).