According to period pants maker Modibodi, transwomen can have 'periods'. Periods which are just as valid apparently.
Whilst I'm not going to suggest I know what hormone treatment tied to a trans woman, I can confidently say that no, they don't have periods. Periods involve blood. Trans women don't bleed.
Modibodi seem to acknowledge that there is no bleeding involved I transwomen's 'periods' but bizarrely under a section on how to manage their periods, trans women should wear period pants. For the life of all God's WHY???! Wtf would wear period pants if they didn't have to?