The Challenger Deep of the Mariana Trench is 11 km deep, and that is just a small fraction of the depth you want to explore (the tallest mountains on Earth are all less than 9 km above sea level), so you'd be three to five times the total elevation range on Earth. This is very, very deep.
A Dredge System Won't Work
Even a 60km to 100km cable or chain reaching from the surface to the bottom would require a lot of material and weigh a lot, and would have to be of a material that would be function at every combination of pressures and temperature from the surface to the bottom. This would be on the same order of magnitude as a cable for a space elevator (low Earth orbit is about 100 km up), and is basically impossible (Pugno 2006) to do so, even with state of the art materials, technology and resources (one estimate cited in the linked paper was $10 billion for an approach that it determined wouldn't come close to working), so there is no way your folks could build that.
What About A Robotic Mining Submarine?
As noted in the comments, there is also no sensible reason to maintain a low pressure part of submarine to retrieve materials from this depth, which makes the engineering problem much greater than one in which an incompressible fluid inside the cavities in the submarine can counterbalance the exterior pressure. But, you would still need to have a control system (either wirelessly remote controlled - possibly with intermediate depth relay buoys, or AI), a power system, a propulsion system, and something to sift through the sediment to distinguish useless material from the materials needed (a non-trival task for materials you can't grab with a magnet).
Mining Sediment Would Be Very Challenging
Any sediment to sift through would tightly packed and might very quicky become sandstone at such high pressures, with a minimal sandy layer, in which case you'd need to break up the sandstone. And, before you got to the sandstone which might have minerals in it, you'd probably have to clear away slushy near ice and organic mud on the bottom. So, you'd need a slush shovel, then a sand mover/sifter, then a sandstone breaker that would make bits fine enough to analyze, and then something to analyze the content.
And, it is entirely possible and indeed, likely that the metals would be tens to thousands of meters under the sea floor, so you might need some underground mining of the sea floor as well to get at the good stuff.
Then, once you segregated out the good stuff, you'd need to bundle it up somehow and have enough propulsion to get it back to the surface (perhaps some sort of compressed air could adjust boyancy and float it up). Also, the good stuff in all likelihood wouldn't be pure elements. It would be, at best, rich ores or oxidates or something like that which would require further processing once you obtained it.
The materials you used would have to not corrode in cold salt water, would have to withstand sometimes strong currents, would have to be big enough to carry an appreciable amount of mined material plus everything you brought to mine it, so even if you unmanned mining sub is barely bigger than a backhoe attached to a small oceanic submarine, you'd probably need something on the order of 10,000-100,000 kg at a minimum. Air independent propulsion rather than nuclear propulsion could probably work, as the total distance travelled wouldn't have to be very great between refueling.
A big investment mining submarine concentrates limited resources in a few high risk projects
But, this is still a major engineering undertaking, and you only get one or two major mistakes and your efforts end up at the bottom on the unthinkably deep ocean. This is a big problem in an environment where there are necessarily lots of unknown unknowns.
What About Filtering Water For Trace Useful Elements?
On the whole, sieving low concentrations of materials dissolved in water (perhaps using a pump to get water from deeper parts of the ocean where mineral concentrations might be greater) seems like a better plan.
For example, on Earth, typical seawater contains the following:
Seawater composition (by mass) (salinity = 3.5%)
Vanadium 1.5 × 10−11 – 3.3 × 10−11
All sorts of goodies are present in sea water in more trace concentrations.
Filtering Water Is Much Simpler And Disperses Risk Across Many Filters
A system that filters impurities from cold high pressure water would have a lot less moving parts and would be easier to set up in multiple small units that could be planted in a line, and could be floated at different depths for different materials.
With a water filtering strategy, it wouldn't be important to land your filters right where rich deposits were found in the sediment the way that it would if you were digging for it. A filter might have five or ten bins each, fitted with balloons that would inflate with something a bit less dense than the surrounding water and float up to the surface (or at least to a manageable retrieval depth like 1-5 km from the surface), that would return collected material as it was gathered. Then, the entire filter system would float itself back up when it ran out of fuel and retrieval bins. A distributed approach would also make occasional failure of a particular filter system or retrieval bin (even say 20%) tolerable in a way that it wouldn't if you put all your eggs in one basket building a massive and complex sea floor mining system where one fail could exhaust all available resources.
The Sea Floor Might Be A Good Place For Simple Filter Systems, Or Not
Indeed, maybe rather than digging in the sediment, you set up an osmosis filter on the sea floor (on the same concept as an industrial strength reverse osmosis seawater filter except that you'd be after the contaminants rather than the filtered water itself) on the theory that mineral concentrations are greater at the interface of sediment and water, rather than actually digging at all.
If you wanted, you could even deliberately fill an area of the seabottom with lots of contaminants by dropping bombs on the sea floor to turn sediment into fine grained debris and then filter out that debris.
But, actually this intuition isn't all that sound. Empirically, all sorts of different trace elements are found at different depths and in different conditions. Lots of elements are most common near the surface, iron seems to be common at mid-depths, and other elements (e.g. lead) tend to favor the bottom of the ocean.
So actually, you'd want filter systems at all sorts of depths after first running probes at different depths and locations to figure out where the elements you want are at the highest concetrations.