Let $A$ be an abelian variety and $X$ a cone over $A$. Then $X$ is log canonical, but as soon as $\dim A\geq 2$, then $X$ is not Cohen-Macaulay. (This you can see by computing the local cohomology at the vertex).
For the $c=1$ case: those are obviously CM. Otherwise one can study the codimensions of various embeddings of abelian varieties.
As ulrich points out, there exist abelian surfaces in $\mathbb P^4$, so those give you $c=2$ with $3$-dimensional singularities.
As any quasi-projective variety $A$ of dimension $d$ maybe embedded in $\mathbb P^{2d+1}$ (Embed in some $\mathbb P^N$ and notice that the closure of the secant variety of $A$ is of dimension $2d+1$, so as long as $N>2d+1$, one may find a point and a projection that gives an embedding of $X$ into $\mathbb P^{N-1}$. Repeat.), you can do that with abelian varieties as well. This gives you $c=d+1$, so turning it around, for any $c>2$ you can find an $X$ that you're looking for in $\mathbb A^n$ with $n=2c-1$.
Now, if you do not need an isolated singularity, then you can take the product of this $X\subset \mathbb A^{2c-1}$ and an arbitrary $\mathbb A^r$, so you get $$ X_{c,r}:=X\times \mathbb A^r\subset \mathbb A^{2c-1+r} $$ of codimension $c$ with $n=2c-1+r$. In other words, yes, you can construct such an $X$ for all $n$ big enough. I am not sure whether the bound $2c-1$ is optimal, but I have a feeling that you can't get much better than that.
If you wanted isolated singularities, I would expect that the codimension is actually increasing with the dimension. In other words, I would expect low codimensional examples in low dimension and not in (arbitrarily) high dimension.
Finally, you do not need to have an abelian variety for this construction. If $A$ is a smooth projective variety of dimension $d$ such that $\omega_A\simeq \mathscr O_A$ and there exist two integers $i,m\in\mathbb Z$ such that $0<i<d$ and $$H^i(A,\mathscr O_A(m))\neq 0,$$ then the cone over $A$ has non-CM log canonical singularities. In other words, to beat the above given bounds you just need to find such subvarieties.
Of course, complete intersections do not satisfy this, but for example the product of any two CYs do. So, you could take, say, a CY hypersurface $H$ and an elliptic curve $E$ and then $A=H\times E$ satisfies to condition (and $A$ is generally neither CY nor abelian).
Unfortunately, the obvious embedding via Segre gives larger codimension than what you get from the above procedure, so this does not obviously give you smaller codimensional examples, but they may be more manageable since you have the product of a hypersurface and a plane curve and even if the codimension is high, you know pretty well the embedding. So, perhaps these are even better examples after all.