There can be VIFs for generalized linear models like logistic regressions. They just shouldn't be calculated the way that the Wikipedia page currently shows, based on $R^2$ for regression of each predictor against all the others. Consider the VIF definition provided by that page:
the quotient of the variance in a model with multiple terms by the variance of a model with one term alone.
The information needed to calculate the VIF is in the variance-covariance matrix of the coefficient estimates. The derivation on that Wikipedia page starts from the equation for the variance-covariance matrix of an ordinary least squares (OLS) model. That then leads to the usual equation for VIF.
With a generalized linear model there is still a coefficient variance-covariance matrix, but it comes from maximum-likelihood fitting rather than a specific analytic form. So the VIF calculation used for OLS wouldn't be appropriate.
vif() function in the R
car package calculates generalized variance inflation factors
interpretable as the inflation in size of the confidence ellipse or ellipsoid for the coefficients of the term in comparison with what would be obtained for orthogonal data.
For an unweighted OLS model this calculation should be equivalent to the usual equation while providing an important generalization for other models. You can inspect the code by loading the
car package and then typing
getAnywhere(vif.default) at the R command prompt.