I don't really know about limousine companies and, if that's a necessary means by which you might travel from the Airport to Miami Beach, have at it!
But just be aware that there is an express bus, number 150, that goes from the Airport bus terminal (accessed by the Airport people mover...just follow the ground transportation signs) to Miami Beach for free (at least for now as it's my understanding that they will resume charging the 2 dollars plus pre-Covid fare as of June 1) . Depending on what hour you get to the Miami Airport will determine how long you wait to depart. The bus ride is 15 to 20 minutes in duration as the Airport is very close to Miami Beach.
In fact, the entire Miami-Dade transportation system is free so hop on and hop off the appropriate bus to get anywhere in Miami Beach and beyond (metrobus, metrorail, and metro mover).
How to learn how to get around? Just access Google maps as you would if you were driving and select the public transportation icon OR download Go Miami into your cell phone OR simply call 311 while you're in the Miami area or call their 305 area code number if you're calling from outside the area.
Just be aware that Google maps makes it appear that you have to take an uber or lyft to get to the Airport bus terminal. You don't as the rolling belts and the people mover train will get you there free of charge in minutes.
And, by the way, that 150 bus has "under the bus" luggage loading capability so no need to lug multiple pieces of luggage into the seating area.
Contact SuperShuttle, they have the contract for shuttle service from the airport.
Alternately, skip the shuttle and just get a cab from the curb outside baggage claim. They are always lined up waiting, and the fare to the Fontainebleau is a flat rate $37+tip for the entire group.