Hi there, Jen!
While there is no official Disney-sanctioned event that happens during the first week of June, crowds are, in general higher than the average simply because it is the start of summer vacation! As more and more children are released from school for the season, families make the trip to
Walt Disney World to enjoy vacation time together.
Many groups (tour groups, sports groups, etc.) plan large getaways to the World throughout the year, but summer is definitely a popular time for the destination. Again, while there is no event that occurs during the first week of June that is officially on Disney's calendar, there are very well-attended events that may affect Park attendance at that time, even more so than the regular uptick of summer traffic.
If you are planning a summer trip, don't forget your hat, your sunscreen, and your water; stay comfortable and safe in the summer sun!
See you later, celebrator,
-Deborah B.