A parent is the only one who can obtain a birth certificate. Your husband will have to do this.
The easiest way (small fee for the document) is to apply to Vital Statistics to get the birth certificate of your husband's child. The other woman does not have to give the birth certificate if she chooses not to.
You haven't explained whose birth certificate. If it's her biological child then she must sign the birth certificate. The biological parents of the child sign the birth certificate.
To get a Florida birth certificate (or a birth certificate in any other state, for that matter), the child had to have been born in Florida.
Birth Certificates are normally handled by the county of birth. You should be able to find the appropriate place to send to on the county web site.
The man on the birth certificate.
Under Illinois state law, you may legally obtain a certified copy of your own birth certificate if of legal age and/or the birth certificate of your child (your name must appear on the child's birth certificate). Court-appointed legal guardians may obtain birth certificates of the children under their care if they submit a certified copy of the appointment. Certified copies of appointments can be obtained from the Cook County Clerk of the Circuit Court. Requests made with legal guardianship papers must be made in person or through the mail.
A child born outside the US will not be issued a US birth certificate. The birth certificate will be issued by the country in which the child was born.
In most places, you would obtain it from a governmental agency that is local to where the child was born. See the related link to the Philippines National Statistics Office.
Contact the Vital Records of the State where the child was born in.
Under Illinois State law, you can obtain a certified copy of your own birth certificate if of legal age and/or the birth certificate of your child. Court-appointed legal guardians may obtain birth certificates of the children under their care if they submit a certified copy of the appointment. Certified copies of appointments can be obtained from the Cook County Clerk of the Circuit Court. Requests made with legal guardianship papers must be made in person or through the mail.
Yes, a child that is legally adopted in the state of Hawaii can receive an Hawaiian Birth Certificate.
Ordinarily this happens shortly after the child's birth.