What is the source? There are multiple mistakes.
"Tribes have gathered honey this way for generations," is a correct and natural phrase.
"Hunting" honey is a vocabulary mistake (hunt = to stalk and then kill animals). "Gathering" honey and "collecting" honey are correct.
"Gathering honey" can be a present participle and object.
"The women and children are gathering honey."
"Gathering honey" can be a gerund and object.
"Gathering honey is one of the duties of the women and children."
"Honey gathering" can be gerund phrase (noun+gerund).
"Honey gathering is one of the duties of the women and children." <<== less natural, but correct
In my opinion, "honey gathering" as a participle of "to honey gather" is wrong.
The women and children are honey gathering. <<== ummm ... no ...
The woman and children honey gather every day. <<== ummm ... no ...
Using the present prefect continous "have been ... hunting" rather than "have hunted" is an odd choice. A repeated action typically uses the present perfect and an incomplete action typically uses the present perfect continous.
Repeated action: John is a house painter. He has painted houses for years. He will paint houses for another five years, and then he will retire.
Incomplete action: John is a house painter. He has been painting my house since Monday. He is going to finish tomorrow.