If you're working with a ridge regression model (as opposed to lasso or elastic net) then its relatively easy to code up a paired bootstrap. With this said and done, (arguably) the standard error for for the coefficients from a ridge model isn't particularly useful. Here's a relevant quote from Goeman
"It is a very natural question to ask for standard errors of regression coefficients or other estimated quantities. In principle such standard errors can easily be calculated, e.g. using the bootstrap. Still, this package deliberately does not provide them. The reason for this is that standard errors are not very meaningful for strongly biased estimates such as arise from penalized estimation methods. Penalized estimation is a procedure that reduces the variance of estimators by introducing substantial bias. The bias of each estimator is therefore a major component of its mean squared error, whereas its variance may contribute only a small part. Unfortunately, in most applications of penalized regression it is impossible to obtain a suciently precise estimate of the bias. Any bootstrap-based calculations can only give an assessment of the variance of the estimates. Reliable estimates of the bias are only available if reliable unbiased estimates are available, which is typically not the case in situations in which penalized estimates are used.
Reporting a standard error of a penalized estimate therefore tells only part of the story. It can give a mistaken impression of great precision, completely ignoring the inaccuracy caused by the bias. It is certainly a mistake to make con fidence statements that are only based on an assessment of the variance of the estimates, such as bootstrap-based confi dence intervals do."
