I guess you refer to the linear discriminant analysis (LDA). In this case, the multinormality of the data is required to guarantee a solution that minimizes the expected error. If you apply the LDA to non-normal data you can not be sure to find the optimal solution. The Fisher discriminant analysis is another popular variant of discriminant analysis (very similar to the LDA) which does not require the normality assumption. FDA (and LDA also) is pretty robust and if your data are not too far off from normality, you will get good results (see here https://www.stat-d.si/mz/mz2.1/sever.pdf).

You would be better advised to use logistic regression. See the attached. Best wishes,David Booth