I went through the documentation and the source code of sandwich's cluster bootstrap function, vcovBS in combination with the coeftest function from lmtest. It looks to me like coeftest does not employ any small sample correction. Is that correct? If not, is there a way to impose a small sample correction with the wild cluster bootstrap & coeftest?
That is correct.
The vcovBS() method for lm objects (the only one supporting wild bootstrap) computes a vector of k coefficient estimates in every one of B bootstrap replications. For the resulting B x k matrix cf it just computes the standard Pearson correlation cov(cf).
If you have a pointer to the small sample corrections that should be applied here, I'm willing to have a look. And if so, would these corrections be specific to wild (cluster) bootstrap or apply to xy or residual (cluster) bootstrap as well?