It depends on each company.
Many Sassie companies operate in the manner you described and ask you to scheduler it for a particular date. For many, you are then expected to do it on that date. Some give you a range of dates and let you choose as long as the report is in by the deadline. If it is ambiguous you can always email the scheduler.
Can you do it outside those listed dates? Maybe. As you gain experience you will learn that schedulers often post shops with earlier than necessary deadlines. For example, today (June 1) you will see shops posted with a date range perhaps from June 1-8th. What if you want one (or more) of those shops but on June 15th and you know the true deadline to submit the reports to that specific client is the end of the month? You can wait until they post it for a date range that includes June 15 and hope no one else takes it or you can ask for it today to perform on the 15th. If you have a track record of success with that company, they will consider your request and might give you that shop today.
When in doubt, do it on the date you promised them. You can learn which shops and companies are more flexible.
Happily shopping Rhode Island and nearby Massachusetts and Connecticut