No, you cannot get around FireRed/LeafGreen's trading restrictions through Pokémon Box. You're required to have completed the Sevii Islands quest to use Pokémon Box in the Kanto titles, which is the same criteria necessary to trade with Hoenn games.
Players must have finished the Sevii Islands quest and activated trades with Hoenn-based games with the Ruby and Sapphire before they can deposit or withdraw any Pokémon.
In every Gen 3 title, you also need 100 Pokémon captured in the Pokedex to withdraw any Pokémon that didn't originate from that save. Emerald requires you to beat the Champion as well, but this isn't necessary in Ruby/Sapphire.
There are several fan-made patches that allow you to circumvent these requirements in Pokémon Box, but I imagine using PkHex or a similar service would be more convenient.
Pokemon Box - Bulbapedia