I have a variable vector to which I know the hard limits of it’s value. It’s true value may be anywhere within that range, it would incorrect to assume it is towards the centre of this range.

I understand for a variable with a uniform prior it’s bounds must be equal to or less permissive than the prior distribution.
Unfortunately, my bounds change for each element of my variable vector.

Declare the vector with bounds of 0 and 1 in the parameters block, map each element to the appropriate bounds in the transformed parameters block, and don’t forget the Jacobian adjustments if you are putting a prior on the thing in the transformed parameters block.

So something like this:
Where h_min and h_max are my bounds

data {
int<lower=0> N;
real<lower=0> C[N];
real h_min[N];
real h_max[N];
}
parameters {
real<lower=0> sigma;
real<lower=0,upper=1> h_sample[N-1];
}
transformed parameters {
real C_hat[N-1];
real h_bar[N-1];
for(t in 1:N-1){
h[t] = (h_sample[t] * (h_max[t] - h_min[t]) + h_min[t]);
C_hat[t] = C[t] * (1/h[t];
}
}
model {
sigma ~ cauchy(0, 1);
h_sample ~ uniform(0, 1);
for(n in 2:N){
C[n] ~ normal(C_hat[n-1], sigma);
}
}

I have not really quite got my head around these Jacobian adjustments. I understand we need to correct for the changes in scale. What exactly should my target += statement look like?

The way you have it written currently, no Jacobian adjustment is needed because you are putting the prior on the parameter h_sample rather than the transformed parameter h. This implies h is distributed uniform between h_max and h_min. Also, you do not need the line h_sample ~ uniform(0,1); because omitting it results in the same log-kernel.