Typically when a student finishes school, they only have approximately 200 to 300 hours and only have a commercial license, not an ATP. An ATP in most countries requires a minimum of 1500 hours.
In the US (since I see that you are in Florida), a charter business is typically operated under part 135 rules. Part 135 rules have the following minimums to act as pilot in command:
PIC requirements:
VFR:
Commercial Pilot with instrument rating or ATP
500 TT, 100 XC, 25 night XC
IFR:
Commercial Pilot with instrument rating or ATP
1200TT, 500 XC, 100 night, 75 instrument (of which 50 are in flight)
There are also other requirements required to operate a part 135 operation, which you can find outlined here:
Federal Aviation Regulations - Index of Part 135: OPERATING REQUIREMENTS: COMMUTER AND ON-DEMAND OPERATIONS AND RULES GOVERNING PERSONS ON BOARD SUCH AIRCRAFT
So, as a short answer... no, someone who recently graduates out of flight school can not just start up a charter company. Well, I guess they could front the money and some brains, find a chief pilot, do the hiring, etc... But a one man charter business operated by a fresh CPL holder is not going to happen.
What a low hour pilot fresh out of school in the USA CAN do (with a flight instructor rating) is start giving flying lessons as an independent instructor, essentially being a one man pilot school.