Using LASSO for variable selection, then using Logit I know this would muddy the statistical inference, but I am really only concerned with getting as close to an accurate model as I can.
I have a dichotomous outcome variable, with a large set of dichotomous predictors. I am thinking I would like to try using LASSO to select which variables I should include in my model, then input those selected variables in to a Logit regression.
Is there something I am overlooking when it comes to the practicality of this approach?
 A: There is a package in R called glmnet that can fit a LASSO logistic model for you! This will be more straightforward than the approach you are considering. More precisely, glmnet is a hybrid between LASSO and Ridge regression but you may set a parameter $\alpha=1$ to do a pure LASSO model. Since you are interested in logistic regression you will set family="binomial".
You can read more here: http://web.stanford.edu/~hastie/glmnet/glmnet_alpha.html#intro
A: First, there's no guarantee that a linear probability model will approximate a logit model very well; consequently the subset of variables selected for one may be less appropriate for the other.
Second, the re-fitting applies no shrinkage at all, despite the variable selection that's taken place in the first step; risking serious mis-calibration & perhaps a little loss of discrimination.
You may be able to validate the procedure on a particular data-set, but it doesn't seem safe in general, or to offer any advantage over a stepwise logistic regression. And of course it's unnecessary; LASSO's $L_1$-norm penalty can be used for shrinkage & selection in logistic regression.
