If the tweets will be posted anytime in the future, you can set up a Twitter Tracker. When started, a new Tracker will gather up to 5,000 tweets from the past 30 days, and continue to gather tweets in real time moving forward. Trackers are available through a Union Metrics subscription, which starts at $99 per month.
If the tweets were posted within the past week, run a one-time snapshot report. A $20 snapshot will include up to 1,500 tweets from the past 7 days, whichever comes first. If you have more than 1,500 tweets, sign up for a paid Union Metrics plan and set up a Tracker, which will start with a backfill of up to 5,000 tweets.
If tweets were posted less than one month ago, then you can access them through a Union Metrics subscription. A new Twitter Tracker will include up to 5,000 tweets from the past month when you start it up. Plans start at $99 per month.