A believable path could be created, but there is no way to justify it in reality
But, then again, why are we here if we're bound only to reality?
I agree completely with L.Dutch's answer. It's hard to see past the efficiency of iron ore.
Here's the problem: the amount of metal bound within a living organism can never exceed the capacity or need of that organism to live its life. As an example, no more iron could be bound within the human body than would allow that body to move to forage, propagate, and live its life. It cannot unreasonably weigh down muscles, it cannot threaten immune systems, it cannot compromise the oxygen cycle or digestion. life, after all, is a delicate balance.
However, people swallow coins and live their lives with bullets in their bodies. Fragments of metal continue in war survivors. I'm not suggesting that any of these general examples suggest it's easy, only that it's plausible for a life form to be believably presented as having a higher metal content.
But, said another way, iron ore will always be a simpler, more efficient way of obtaining iron than farming a biological factor no matter how well crafted to maximize its metal-carrying abilities. The density of iron ore will always exceed the density of metal in living organisms by many orders of magnitude.
But that's boring
I could suspend my disbelief to enjoy a story that required farming living organisms for metal so long as the substantial inefficiency involved was justified. IMO, it's not enough to rationalize higher metal content in living organisms...
...you also have to rationalize why it doesn't exist in mineable quantities — and yet exists in sufficient dispersed quantities — such that it makes sense to farm animals at all. Then, you need to explain why evolution allowed all that metal to bond to living creatures in the first place.
Let's chase that and see where it leads us
Disclaimer: Biology is NOT my strong suit. If someone practiced in the art reads this and find themselves capable of typing a comment to point out my error amidst their convulsions of laughter, I'd be grateful.
Lactoferrin is a nutrient classically found in mammalian milk. It binds iron and is transferred via a variety of receptors into and between cells, serum, bile, and cerebrospinal fluid. It has important immunological properties, and is both antibacterial and antiviral. In particular, there is evidence that it can bind to at least some of the receptors used by coronaviruses and thereby block their entry. (Source)
We have a world that has a lot of iron, but for some reason, that iron is never found in a chunk bigger than the tip of one's finger. It's thoroughly dispersed through the soil, where it can easily be picked up by plants. Plants will draw almost anything from the soil, but I want to jump a step further and justify that extra iron in an animal.
The quote above explains that there is a mammalian protein that binds iron: Lactoferrin. The animal will consume iron via the plants thanks to the high concentration of what we'll call molecular iron in the soil. The lactoferrin gets it into the body and has the potential of keeping it there.
As L.Dutch said, evolution is a harsh mistress and an even harsher accountant. For example, in humans too much iron can compromise zinc absorption, and that compromises the immune system. But that same link explains that we humans need iron to produce hemoglobin, a protein in red blood cells.
Cool, so why would life on your planet need higher concentrations of hemoglobin? Among its uses, it carries oxygen to cells and carries away carbon-dioxide.
Which means, using suspension-of-disbelief as the criteria, you could suggest that your planet has a higher percentage of carbon-dioxide in its atmosphere, requiring a greater quantity of hemoglobin to better carry in the limited oxygen and carry out the too-abundant CO2, which means more iron is needed... and we have Lectoferrin as one possible rationalization for how to do that.
And as mentioned earlier, it's all because on your planet there are no iron deposits anywhere. Perhaps (and I'm skipping to a simple explanation because I'm out of time to write this answer), volcanism is long-past on your world, and erosion has scattered the iron and bound it with oxygen (helping to rationalize the limited oxygen supply). In other words, your world has rust dust everywhere.
But no iron mines anywhere....
I believe you can rationalize higher metal content in living organisms such that they could be mined for metal. But even with the increases you're suggesting, it's an horrifically inefficient process. This suggests that you need to further rationalize why no iron mines can exist on your world — and binding the two together as cause-and-effect can solve your problem.