Some people (usually those who want more control, more optional rules, or just don't want to surrender dice rolling to an RNG) do the 'battle' parts of the game over TableTop Simulator (TTS) instead of MegaMek, too. And there are simpler 'shared view' map tools, as well.
And of course, there's also the option of using Discord for voice/video chat, either 'theatre of the mind' or with Roll20/Foundry/etc providing the map.
Play by post would be possible, but also very lengthy, at least as far as using BT 'mech combat rules goes.