Typically schools are required to enroll children who qualify for special education services in those services, which is a federal requirement, so that particular law should be the same in FL as it is NY. However, speech services would most likely involve a certain number of minutes per week working with a speech teacher - not a full- or even half-time enrollment in a program that would also serve as childcare.

In terms of the speech services, I would find the number for the department of special education (or director of speech/language pathology) in the district you are in. Someone in that office should be able to direct you to the appropriate next steps in terms related to securing services.

In terms of childcare, my guess is that those services wouldn't be long enough to provide you enough time to get to work and back. I do know that, in many areas, child care vouchers/subsidies are sometimes available, but often require the family to qualify through the department of social services or human services, and there is often a waiting list for such vouchers/subsidies, especially with the current economic climate.

I wish I knew more about childcare for kids that young, but unfortunately sometimes it can be really hard to come by, and is a real struggle for families. It may be helpful to just start asking around if anyone knows of any resources available, such as through community agencies or churches. If you talk to someone and they don't have anything available, it may be helpful to ask them if they know anyone else you should talk to.

Being 3 years old, I'm not sure that he would qualify for programs through a school - he would need to be enrolled in pre-k.  I live in Virginia and have a son in speech therapy.  At about your son's age, I asked my pediatrician about speech delay and he thought it wasn't severe enough to be concerned and advised me to wait until around 5-6 years old to have him evaluated.  In kindergarten, my son was tested by the special education department for it, but he placed just around the lower end of his age group so he did not qualify; yet the following year he tested lower and qualified, and so through the special ed dept he now receives speech therapy once a week during regular school hours.  So, first, I would ask a pediatrician or two for an opinion on the severity of speech delay for your child and possible solutions; if it's not too severe, wait for him to be old enough for school and contact the special education resources coordinator.  In Virginia, and I'm sure Florida is probably similar, there is a long list of rights for children that are eligible for these types of resources, and it shouldn't cost you anything.

My son began getting services through the school district at three years old throught he Early Childhood Special Education Program (ECSE).  If there is a special need they can begin at three years old in our state.  I do not know for sure about your state.  He received speech therapy and occupational therapy and went to school for three and half hours a day.  Once kindergarden starts you can develop an IEP which would give him a certain amount of therapy each week.  Good luck!  Definitely talk to the school district or the education board for more information about your state.

Most areas will have at least one center that is specifically designed to work with 3-6 year old children in a preschool setting. You can call your school county office to find out more about this department.
Make sure you request everything in writing and keep copies. They have 90 days to work with you.