I am aware it is common practice to standardize the features for ridge and lasso regression, however, would it ever be more practical to normalize the features on a (0,1) scale as an alternative to z-score standardization for these regression methods?
If you apply normalisation (squeeze in [0,1]) you will have a measure of relative variable importance but it will change the scale of your variables and you will lose all model interpretability. The advantage of standardisation is that you can still interpret the model as you would with unregularised OLS regression (this has already been already answered here).
Normalization is very important for methods with regularization. This is because the scale of the variables affect the how much regularization will be applies to specific variable.
For example, suppose one variable is in a very large scale, say order of millions and another variable is from 0 to 1. Then, we can think the regularization will have little effect on first variable.
As well as we do normalization, normalize it to 0 to 1 or standardize the features does not matter too much.