They need to be able to breathe, so an airlock leading into a section of ocean wouldn't be useful
Well, not so much. Freediving is very much a thing, after all, and has been for quite a considerable length of time. People have been able to variously dive to over 100m, swim hundreds of metres and hold their breath for about 10 minutes (probably not all three at once, mind you). You can look back centuries to find traditions of people diving without breathing gear to recover wrecks, pearls or food. Plenty of interesting stuff out there... have a read about ama divers, for example.
Once you're out in the water, spearfishing is a reasonable technique, and one often used in combination with free diving.
So all you really need is a moon pool or fast cycling airlock and a bunch of people willing to practise apnoea.
Now, the issues of diving illness and long term human viability at depth, that's a different matter...
Edit to add some alternative food sources
Remember that whilst fishing can be a very effective source of nutrition, it needn't be the only game in town and alternatives may be easier to arrange.
Shellfish are one good possibility... oysters and mussels are already farmed commercially. They have potentially useful byproducts (shells, pearls) and perform useful ecological services such as removing carbonates from the water and filtering out nutrients that might otherwise cause problems (like poop from fish farming). Mussels can be farmed on a rope, and the whole lot hauled in for harvesting without anyone having to take a dip.
Many varieties of seaweed and other algae can also be farmed if you are high enough in the photic zone for them to thrive. Not just useful for food, but you can use them to make other handy raw materials like fuel or plastic feedstocks. In either case, you don't necessarily need constant hands-on work to get your food, making the logistics of the whole thing much simpler.