Initially I wondered why you wanted to derive the full conditional for $\sigma$. But in the comments you say:

> My research adviser, in preparation for working with him, has given me the task to compare the *a priori* inverse-gamma with the log normal stated in the post.

To this end, would it suffice to simply have the two analytical forms of the priors?  Perhaps the end goal is actually comparing the posteriors though.  Your adviser also said

> ...it is possible to implement the [model to determine the] posterior with an MCMC algorithm

which is true.  In which case, I suggesting working with the full-conditional for $\log\sigma$ since this is how you specify it in the model.  Once you have a posterior estimate for $\log\sigma$, it is then trivial to get a posterior estimate fro $\sigma$ which you can then compare to your results from your atlernate model with the inverse-gamma prior.

Assuming that $\mu_0$, $\sigma_0^2$, $\mu_\sigma$, and $\tau_\sigma^2$ are known *a priori* (since you don't give them priors), then the full conditional for $\log\sigma$ is:

$$ \begin{align} 
p(\log\sigma | \boldsymbol{y}, \mu) & = p(\log\sigma) \times p(\boldsymbol{y} | \mu, \sigma^2) \\ 
& = \frac{1}{\sqrt{2\pi \tau_\sigma^2}}\exp\left(-\frac{1}{2}\frac{(\log\sigma-\mu_\sigma )^2}{\tau_\sigma^2} \right) \times \prod_{i=1}^n \frac{1}{\sqrt{2\pi \sigma^2}}\exp\left(-\frac{1}{2}\frac{(y_i-\mu )^2}{\sigma^2} \right) \\
& \propto \exp\left(-\frac{1}{2}\frac{(\log\sigma-\mu_\sigma )^2}{\tau_\sigma^2} \right) \times \frac{1}{\sigma^n} \exp\left(-\frac{1}{2}\sum_{i=1}^n\frac{(y_i-\mu )^2}{\sigma^2} \right) \\
& = \frac{1}{\sigma^n}\exp\left[-\frac{1}{2}\left(\frac{(\log\sigma-\mu_\sigma )^2}{\tau_\sigma^2} +\frac{\sum_{i=1}^n(y_i-\mu )^2}{\sigma^2}\right)\right]
\end{align} $$

Together with the other two full conditionals, you could implement this using an MCMC algorithm (e.g. Metropolis Hastings, adaptive rejection sampling, Hamiltonian Monte Carlo, slice sampling, etc.)

If you are only interested in obtaining an MCMC posterior estimate, then you could implement this in software like BUGS or JAGS which doesn't even require you to determine the full conditionals.