I know there are some free clinics that don't ask for ID, at least in the city here (I just read a book about street kids in NYC and they talked about how they never had ID cards and lost it along the way but there were free clinics who would take them in and youth centers with doctors that would check them out).
Maybe your area has a clinic like that?
If not, I would get your ID right away. If you have had an ID card in the past, go to your DMV and tell them you lost it. They'll print you out a new one that day.
Also, if you go to a doctor and explain you don't have an ID card but you have another form of ID such as a birth certificate or SS card, he may take you.
I'm not sure if you are pregnant or not, but either way you should probably see a doctor because not having a period for 3 months could be a sign of another problem if you aren't pregnant.
Good luck!