Lawfully present immigrants are eligible to enroll in Obamacare.
Lawfully present would be: “Qualified non-citizen” immigration status without a waiting period.
https://www.healthcare.gov/immigrants/immigration-status/
In Florida you can enroll during Open Enrollment which is Nov 1-Dec. 31.
If you have low income your premium would be subsidized, and you may qualify for cost sharing. https://www.healthcare.gov/lower-costs/save-on-out-of-pocket-costs/ This will help you pay for you and your families premium and drive down the deductibles, co-payments and out of pocket max.
If you had a special life event within the last 60 days, you may qualify for a Special Enrollment. https://www.healthcare.gov/coverage-outside-open-enrollment/special-enrollment-period/
You can enroll in Medicaid and Chip throughout the year whether you have a special enrollment or not. https://www.healthcare.gov/medicaid-chip/getting-medicaid-chip/