EDIT: The validity of this approach depends on whether your next proposal begins with the rounded values or the non-rounded values.
New answer assuming that your next proposal uses the non-rounded values
Yes, this approach is valid. It can be formalized as follows. In place of a likelihood $L(X | \theta )$ and prior $\pi_\theta(\theta)$ where $\theta$ is discrete, define a new likelihood $Pr(X | \phi) \equiv Pr(X|R(\phi))$, where $\phi$ is continuous and $R$ rounds every entry to the nearest valid value. Run your sampler on $\phi$.
N.B. Some people might have the intuition that the prior on $\theta$ corresponding to the above scheme is to evaluate $\pi_\phi$ at $\theta$. It's not. A given prior $\pi_\phi(\phi)$ induces the following prior on $\theta$: $$\pi_\theta(\theta) = \int_{[\theta \pm 0.5]} \pi_\phi(z)dz$$.
Old answer assuming that your next proposal begins with the rounded values
This is not a valid approach. This affine-invariant sampler is a Metropolis-Hastings variant, which means the proposal distribution appears in two different spots in the algorithm. It appears in generating proposals, of course, and you've altered that spot. But the PDF of the proposal also is used when deciding whether to accept or reject the current proposal. If you change it in one place and not the other, there's no guarantee what will happen.
(EDIT: Even if you were to evaluate the density of the proposal correctly, it may break the symmetry property $g(1/z) = zg(z)$ that the sampler relies upon to achieve detailed balance.)
There's another issue here: the "stretch move" proposal used by this affine-invariant sampler uses a discrete proposal distribution! Each walker selects another walker and moves a certain percentage towards or away from that other walker. The percentage is fixed; it's a tuning parameter of the algorithm. I can't say for a fact that this is your problem, but it's easy to imagine a scenario where the step size is small, and your rounding drags it back to the current value no matter what. Could this be causing your problem? Yes.
You may be wondering what to do next, but that's a matter for another question:
What MCMC algorithms/techniques are used for discrete parameters?