There are several alliance guilds that are social-oriented but also raid. The ones that I can think of off the top of my head are Pizza Warriors (Americas), Four Words Guild, and the Cult of Dionysus. There are many others too.
However, whether you get to actually raid with them if you do manage to join is not a guarantee. As you can imagine, there are a lot more casual players than there are hardcore raiders and the demand for raids at the casual level can be higher than the supply i.e. the number of raids accepting low geared/experienced players. If I were you, I'd try to reach out to these guilds and inquire further -- they tend to be friendly and open.
Also, despite you and your wife playing very valuable roles in theory, I'm afraid to tell you that in practice, the specs the both of you chose will make getting into raids quite a bit harder. People on Lordaeron tend to follow the meta very strictly so:
- It is very unlikely for raid leaders to take prot warriors as they are deemed inferior to DKs and paladins. They only time they are in demand are for very niche uses in endgame raids i.e. tanking adds in TOGC on the last boss and RS 25 hc.
- Fury warriors are definitely a meta spec but literally everyone in this realm has a fury warrior, therefore you will have to compete HEAVILY for a spot. You will notice in 9 out of 10 raids, fury warrior and dps DK spots get filled almost immediately. It's not impossible, but again, you'll have to be fast and aggressive to secure a spot compared to other specs.
- Disc priest is also a meta spec but unfortunately there is only room for ONE in any given raid (and they are usually only wanted for 25 man raids). This is due to the way the Weakened Soul debuff works, making additional disc priests redundant.
Personally, I'd recommend you do the following:
- Focus on playing your fury spec going forward. Even if you do prefer to play prot, you'll have an easier time getting gear for your prot spec as a fury warrior than because you'll be able to actually get into raids and roll for gear. Tank gear is not very contested so you could gear up entirely from off-spec (OS) rolls that way.
- Your wife could consider switching to a shadow spec until a more permanent opportunity opens up to play disc. Shadow priests are more stackable in raids and isn't too hard to play. Again, if she prefers to play disc, she could gear up that spec by rolling for OS.
- Consider making your own raids. I know this step tends to make people uncomfortable if they don't like leading -- and it definitely was for me too -- but this is one of the surest way for you to guarantee you and your wife get a spot in a raid. Start small and advertise your runs as learning runs on global chat. People tend to be more understanding that way and you might even get experienced/geared players to come help and offer advice.
Good luck on your journey!