Is this company required to file taxes for the previous year, and if
they don't, will they face any punishment?
Yes. Every corporation is required to file an IRS Form 1120 every year (the same rule does not apply to trusts or limited liability companies), even if it has no revenue or expenses. See 26 USC § 6012(a)(2) and Treas. Reg. § 1.6012-2 ("Except as provided in paragraphs (e) and (g)(1) of this section with respect to charitable and other organizations having unrelated business income and to certain foreign corporations, respectively, every corporation, as defined in section 7701(a)(3), subject to taxation under subtitle A of the Code shall make a return of income regardless of whether it has taxable income or regardless of the amount of its gross income.").
The penalty, assuming that no tax is actually due as per the question, is a modest fail to file a return fine that was $210 in 2018. 26 USC § 6651(a) and (i); Rev. Proc. 2018-18. The fine will be $215 in 2019 ($205 for an S-corporation). Rev. Proc. 2018-57.
A state income tax must be filed by a Florida corporation only if it has income in the state in a given year. So, while there is a federal requirement and penalty, there is not a state one.