Publicly-available Twitter datasets (such as those listed in the catalogs below) will contain lists of Tweet IDs. You will need a utility (such as Documenting the Now's Tweet Hyrdator) to retrieve the full JSON files of the Tweets.
There are a variety of third-party (for-profit) entities that can create custom Twitter datasets based on your specifications. Read reviews and shop around carefully; this method is fairly expensive.
Other researchers may also be willing to share Twitter data they've collected for their own work for other academics to reuse.
