Yes

and

If you think about it, what else could it be using?

Write $\hat{\beta }$ for the svyglm estimator. Theory says the estimator solves the weighted score equations

$$U\left(\beta \right)=\sum _{i=1}^N\frac{R_i}{{\pi }_i}{U}_i\left(\beta \right)=0$$

where $N$ is the population size, $R_i$ is the sampling indicator, and $U_i={\partial }_{\beta }{\ell }_i\left(\beta \right)$ is the score. Doing an Taylor series expansion on this gives

$$U\left(\beta \right)=U\left(\hat{\beta }\right)+(\hat{\beta }-\beta )\frac{\partial U}{\partial \beta }+\text{remainder}$$ so that $$\hat{\beta }-\beta \approx {\left[\frac{\partial U}{\partial \beta }\right]}^{-1}U\left({\beta }_0\right).$$ The large-sample variance approximation is then the ‘sandwich’ $$\widehat{\text{var}}\left[\hat{\beta }\right]={\left[\frac{\partial U}{\partial \beta }\right]}^{-1}\widehat{\text{var}}\left[U\right({\beta }_0\left)\right]{\left[\frac{\partial U}{\partial \beta }\right]}^{-1}.$$

This is all similar to, eg, Huber or White’s derivation of the sandwich estimator. The only difference is that the middle term¹ has to be estimated differently because of the survey design. That is, the svyglm variance estimator generalises the familiar sandwich estimators to allow for non-trivial sampling.

The middle term is the variance of an estimated population total, and is estimated the same way as for any other population total. This is literally true: all the population-total variance estimates go through the function svyrecvar.²

The middle term is $$\sum _{i,j}\frac{R_{ij}}{{\pi }_{ij}}\frac{R_i{U}_i}{{\pi }_i}\frac{R_j{U}_j}{{\pi }_j}$$ where ${\pi }_{ij}$ are the pairwise sampling probabilities. If you had independent sampling of individual records, so $\text{cov}[R_i,R_j]=0$, the middle term would reduce to $$\sum _i\frac{R_i}{{\pi }_i}{\left[\frac{R_i{U}_i}{{\pi }_i}\right]}^{\otimes 2}$$ and the whole thing simplifies to a standard sandwich estimator.

Model-based standard error estimates are based on simplifying the sandwich estimator by making stronger assumptions about the structure of the middle term. We can’t do this with survey data: we don’t necessarily assume anything about how the finite population was generated, so no simplifications are available.

So, yes, all the model in the survey and svyVGAM packages use model-robust standard errors.