For one thing, regardless of the state, if someone refuses to have his conversation recorded, you cannot force him to speak. In Florida, in some cases it would be legal to record the conversation without the consent of the other party. In some (perhaps most) cases it is not legal.
Florida's recording statutes have some weird exceptions. The recording statutes are covered in Chapter 934.
The consent of all parties is required for wire, oral and electronic recordings. To record and disclose the contents of recordings can be, in Florida, either a misdemeanor for a first offense where there was no illegal purpose or commercial gain in the recording, or a felony. Florida Stat Ch 934.03.
Interception of calls in which one is not a party violates both federal and Florida law.
For the non-electronic taping of a person who has no expectation of privacy, no consent is required. An example of such a legal recording would be of a public meeting. Fla Stat Ch 934.02.
It is also legal in Florida for businesses that use telephones during the regular course of business to record business-related telephone conversations on a business extension. This is supported by caselaw. Fla Stat Ch 934.10.
So, the licensed child care provider cannot record a private conversation with the state DCF employee without the consent of the DCF employee. If the DCF employee is speaking in public, where he has no expectation of privacy, however, then his consent would not be required.