Can I walk to a specific biome with Baritone?

Baritone is not omniscient. It's just a client-side mod, and doesn't know about the world any more than the server is willing to share. It does exploit some info the server provides even though it wouldn't need to for the client to work, like location of nearby ores, but if the server didn't provide location of a mushroom biome, or the seed, Baritone has no clue where it could be.

If you have the world seed, use a biome finder service to find a mushroom island or a roofed forest, and go there, launch Baritone when you arrive. If you don't, you may attempt to crack it; it's not easy but it's possible. Don't expect the poor bot to be omniscient.

On the other hand, you can let Baritone #explore for a couple hour, then tell it to #mine mushroom_stem red_mushroom_block brown_mushroom_block and see if it found any in its travels.