I can't speak for Indiana but "Can they legally do that"? seems to be the entire test in the case of the new Florida law. That is exactly the law the legislature passed, that a business MAY NOT stop an employee with a permit from keeping his firearm locked in his car on company property.
Now that has resulted in a court ruling upholding the law for now so at the moment a business may not keep an employee with a permit from having a gun locked in his car. HOWEVER, the court says since the law specifies an employee, that it therein means that they MAY keep non-employees (spelled US) from having guns in their cars. Of course that was never the intent and hopefully will be corrected quickly.
The battle is between property rights of the business and the individual rights of the permit holder. It appears so far that even though the legislature has spoken to give the upper position to the individual the courts are trying to doubletalk their way back to the business.
How will it all turn out??
If Indiana does not have a specific law addressing the issue, then property rights may well be paramount and the answer may be YES.
The Indiana Senate has a bill before right now that, if passed, will resolve this issue in our favor.
Firearms in locked vehicles. Prohibits a person from adopting or enforcing a policy or rule that prohibits or has the effect of prohibiting an individual from legally possessing a firearm that is locked in the individual's vehicle while the vehicle is in or on the person's property. Excepts possession of a firearm: (1) on school property or a school bus; (2) on certain child care and shelter facility property; (3) on penal facility property; and (4) in violation of federal law. Provides that a person who, in compliance with the prohibition, does not adopt or enforce such a policy or rule is not liable for resulting injury or damage. Authorizes a civil action for damages, costs, attorney's fees, and injunctive relief to remedy a violation.