Carla, Hardee County was formed in 1921 from Desoto Co. However you probably won't find a birth record. Check out the Florida Bureau of Vital Statistics web page at
http://www.doh.state.fl.us/ for information about the dates Florida began keeping birth and death records in a centralized location.
My grandmother did not have a birth certificate, but contacted the school she attended to obtain a certified letter showing her date of birth as shown on their records as well as her parents names.
If your grandmother had a social security number, you may want to contact the Social Security Administration to obtain a copy of her original application for a social security number. The cost is $7.00 if you have the social security number.
Have you checked to see if there was a family Bible? If so, you will want to make a copy or record the front page that shows the publisher and publication date as well as recording the births, deaths, marriages in the Bible.
Good luck with your search. Teresa