Using Neural Network for locating shifting resonant frequencies
Is it possible for a neural network to 'detect'/'pick out' these frequency values?

Yes, that sounds generally possible.
There's the Universal Approximation Theorem that says that a sufficiently large Neural network¹ can approximate any continuous function on anything isomorphic to $\mathbb R^N$ (and your FFT output is that), including things like

map this FFT output to a vector of resonant frequencies.

Now, what that says that if such a continuous function exists, then a neural network can reproduce the same.
It's then &quot;just&quot; a problem of training the weights of that neural network. That training usually happens through putting in data and judging the output (e.g. by how &quot;correct&quot; the output is, which requires you to already know the correct output, or by making sure the output probability distribution is as close as possible to what the distribution of the true phenomenon is).
However, whether or not that function exists is more of a physical / stochastic problem: For example, when I want a function

map the FFT of the acellerometer signal to the weight of the fattest cat of the oldest cat-owning person on this planet

no neural network could approximate that – simply because there's no relationship between the signal and the cat.
So, a Neural Network can't do magic: if things are stochastically independent, or ambiguous, then no estimation can be implemented, by a Neural Network or anything else. That's kind of an important thing to keep in mind, especially when reading the hype literature about Neural Networks that still occasionally show up:

If there's no mutual information between in- and output of the proposed mapping, then there's nothing any algorithm could do – be it a Neural Network, or anything else.

But in your case, I think you have good reason to assume there's a solid amount of relation between your input data and the output you want, so you should be fine.

¹ &quot;sufficient&quot; can be very large; and you need activation functions that aren't polynomials; what we call &quot;neural networks&quot; usually doesn't use polynomials.