Harry goes to 12 Grimmauld Place after the dementor incident to find that the Weasley children and Hermione have been living there along with the Order. From my understanding, the twins, Ron, and Ginny have to be there because their parents are doing Order work, but Hermione's family obviously isn't. And since children aren't allowed to be official Order of the Phoenix members and be let in on things, there doesn't seem to be a reason for Hermione to be there. Is there a reason why she is present there?
She was spending the rest of her summer vacations with the Weasleys, as she did the previous summer
Grimmauld Place was not serving only as the Order's headquarters, but as the Weasleys' temporary home as well, in that particular timeframe.
Don't forget that Harry was neither a member of the Order, yet he went to the Order's headquarters as well to spend his summer vacations. Both he and Hermione went to the place where Ron and his family were staying, to spend their time with their friend -they were not being recruited to a secret brotherhood.
We don't have an exact date on when The Weasleys moved to 12, Grimmauld Place, but that probably happened enough time before Harry arrived there:
“Thanks to you, Dumbledore was able to recall the Order of the Phoenix about an hour after Voldemort returned” said Sirius.
Harry Potter and the Order of the Phoenix, Chapter 5 - The Order of the Phoenix
Even though we can assume that Dumbledore didn't select Sirius's house that early, the period between that moment and the time Harry arrives is around a month -plenty of time for the Weasleys to move in and for Hermione to join them.