You can research each state you will be driving through to see their laws and if they allow carry without a permit in the car.
For instance if you click on Florida an scroll down you get:
RV/Car Carry Without a Permit/License
790.25 Lawful Ownership, Possession, and Use of Firearms and Other Weapons
(5) Possession in Private Conveyance.--Notwithstanding subsection (2), it is lawful and is not a violation of s. 790.01 for a person 18 years of age or older to possess a concealed firearm or other weapon for self- defense or other lawful purpose within the interior of a private conveyance, without a license, if the firearm or other weapon is securely encased or is otherwise not readily accessible for immediate use. Nothing herein contained prohibits the carrying of a legal firearm other than a handgun anywhere in a private conveyance when such firearm is being carried for a lawful use. Nothing herein contained shall be construed to authorize the carrying of a concealed firearm or other weapon on the person. This subsection shall be liberally construed in favor of the lawful use, ownership, and possession of firearms and other weapons, including lawful self-defense as provided in s. 776.012.
790.001 Definitions.-- As Used in This Chapter, Except Where the Context Otherwise Requires:
(17) "Securely encased" means in a glove compartment, whether or not locked; snapped in a holster; in a gun case, whether or not locked; in a zippered gun case; or in a closed box or container which requires a lid or cover to be opened for access.
790.25 Lawful ownership, possession, and use of firearms and other weapons.—
(1) DECLARATION OF POLICY.—The Legislature finds as a matter of public policy and fact that it is necessary to promote firearms safety and to curb and prevent the use of firearms and other weapons in crime and by incompetent persons without prohibiting the lawful use in defense of life, home, and property, and the use by United States or state military organizations, and as otherwise now authorized by law, including the right to use and own firearms for target practice and marksmanship on target practice ranges or other lawful places, and lawful hunting and other lawful purposes.