In the United States, we'd just say "mushroom hunters." To tell the truth, the people I know who used to go looking for mushrooms didn't even say that--they'd use the name of the particular mushroom and say "were going to look for morels." With many hobbies, there is a kind of dual vocabulary, in which there is an official name for the hobby that is very formal and high in "register," but the people who actually practice the hobby call it by some informal, simple name. I don't think anyone would say "I'm a philatelist" with a straight face; they'd say "I'm a stamp collector." Even something as simple as "birdwatching" gets shortened; all the birdwatchers I know call themselves "birders."

Mushroom hunters. Or, rarely, mushroomers

In my area, we just call them mushroom hunters.

Mycophagists, mycophiliophagists, or mycogastronomes. These are scientific terms that most people don't know. you can simply say mushroom pickers / farmers (since a lot of people don't exactly know what do farmers pick up whether it's herbs, plants or mushrooms..)

People who like picking up mushrooms are mushroomers) or habitual gatherer of mushrooms)