The verb I've most commonly heard for this activity in English is "forage". What's the equivalent German word?

"Mushroom hunting" is a fairly common phrase in English, too. It appears to have the top-level title for the page about that activity, on Wikipedia, even (mushroom foraging, mushroom picking, and mushrooming are all given as alternative terms)

Plus it's the title of a song on the Cowboy Bebop soundtrack, so it has that going for it.

Thank you for clarifying!

The German term is "Pilze sammeln" which literally translates to: collect mushrooms.

There are many dialects of the German language - where I'm from, we would use "Schwammerl suchen" ("Schwammerl" as another term for "Pilz(e)"). This literally translates to: searching for mushrooms.