Is there away of determining how many and which users were active on a given day?
Without of course visiting the individual profile pages.
You have to define active more precise but I have taken the users that either post or edit or leave a comment and then created this SEDE Query
select activedate , count(*) [# users] from ( select distinct cast(creationdate as date) as [activedate] , owneruserid from ( select creationdate , owneruserid from posts union select lasteditdate , lasteditoruserid from posts union select creationdate , userid from comments ) data ) uniquerows group by activedate order by activedate
You can extend this query to also include events from the posthistory table and suggested edits. See Database schema documentation for the public data dump and SEDE for the possibilities.
When run today this is what the graph looks like:
Keep in mind SEDE is only updated once a week, on Sunday. If you have more questions that can be answered with queries against the SEDE database, consider asking them on Meta Stack Exchange. They already have a bunch of queries ready and if none of the existing ones fulfill your need there are enough users around that are prepared to take on these query requests.