There are many festivals in June. Here are the ones I collected. Take a look. However, the only festivals that require you to send gifts seem to be Children's Day and Father's Day.
June 1 International Children's Day June 5 World Environment Day June 6 Grain in Ear June 11 China Population Day June 14 World Blood Donor Day June 17 Father's Day (the third Sunday in June) June 19 Dragon Boat Festival (the fifth day of May) June 20 World Refugee Day June 22 Summer Solstice June 23 International Olympic Day June 26 International Day Against Drug Abuse (International Anti-Drug Day) June 30 World Youth Festival
For Children's Day, just give children dolls, dolls, chocolates, toys, etc. that they like; for Father's Day, just give some health products, watches, clothes, digital products, and other things that middle-aged and elderly people like. If you are still worried, it is recommended that you find some more professional gift websites and consult those more professional gift experts.