In the case of Mother Teresa, just a few months lapsed between her death and the opening of her cause. After it gets into the bishop's hands, it's out of the group's. There's still the matter of finances. The group can't buy a canonization, but getting someone canonized takes money. Expenses have to be paid.
Research has to be done.
Someone has to oversee the whole project. Results need to be collected for presentation to the Vatican. That's another advantage of the grassroots support. Not only can the group ask for prayers for the candidate's cause but it can ask folks to send funds to help cover costs. That's important because while the case moves from the diocese to Rome, someone has to foot the bill.