Let me preface this by saying I do not have particular expertise in either logistic regression or working with panel data. That being said... I am not aware of any technical limitation on using logit + panel data. PROC LOGISTIC has a CLASS statement, so you could use your date variable in that statement or just as a continuous independent variable.
You should be okay with nominal/unordered data. Per SAS documentation
For nominal response logistic models, where the possible responses have no natural ordering, the logit model can also be extended to a multinomial model known as a generalized or baseline-category logit model
The part I'm least familiar with, this being panel data and potentially needing to track each person, I googled 'sas logit fixed effects' and found this post which mentions using PROC SURVEYLOGISTIC with the STRATA statement. That may be your best next stepl