No. It is not necessary to purchase the Pokémon HOME premium plan. However, it is required to charge energy in order order to Transport from Pokémon GO. The energy used to transport Pokémon is fully charged at first, and after use will recharge over time. Additionally, it is possible to charge energy without waiting by using Pokécoins. Information on purchasing Pokécoins can be found here.
*'Here' links to Niantic's page on Pokécoins.