You're right, they do call forms "Business Process" and also certain type of workflows "Business Process" and that throws me every single time. You used the right terminology, I just didn't pay enough attention to which type of "Business Process" you were referring to, so that was totally my bad.
Based on the way you have described it, if I were making this process, I wouldn't do it as a process that the staff start themselves. I would make it a scheduled process that assigns the task to the full team of users. If you are on version 11 Update 2 or Update 3, you can also assign it by group instead of user or team. Then any one of the users included in the assignment can claim the task and complete it. Include the Timer Catch Event to interrupt the task and finish it if it isn't completed by a staff member before end of day. This way, it's only starting once, because you have the schedule starting it, not the staff. It's only being completed once, because someone has to claim it and complete it, and it can't be claimed and completed more than once. And if no one completes it, the system still finalizes it, so that you have the record that it wasn't done that day.
If you really do want users to start it, then you'll need to have a lookup to identify if it has already been processed for the day, and use that in conjunction with Field Rules or similar limitation to ensure it cannot be started again (hide the submit button for example). You could make a simple View on your LFForm database to report the max start timestamp of this particular process - then the lookup returns that value and you have some formulas or Javascript to indicate that the the form cannot be processed if that date is too recent. But this has a flaw if done as a Message Start Event that if two people are in the submission form at once, the lookup won't catch each other, and then they can both submit. So instead, you'd need to do a Start Event instead of a Message Start Event, then run a workflow to check the history, and report back to forms on a user task. This should work, but it does feel more complicated than the schedule option in my opinion.