Well, you *COULD* always OVERWRITE the file in the shared folder. So the file might grow, but there could be only one file, if you wished.
Your instinct to use a database is MUCH MUCH BETTER IDEA!!! Good for you!
Now you WOULD use IIS on a webserver. And yes, you COULD use ASP code (written in VBScript, so you don't have to learn another language right now) to do this.
You could save the data in one or more tables in a SQL Server or MySQL or Access database (in order of choice). You could have a field in the table(s) that tells who owns the particular data, so that when a user comes back to the page you could, if they wanted you to, re-load the data from the DB. More than that, any administrator with proper permissions could look at ANY of the records in the table(s) and so view what any person in the company has saved, if that is at all desirable.
If your boss insists, then yes, you could do this all in the browser. But your idea is, I think, at least 10 times better.
An optimist sees the glass as half full.
A pessimist sees the glass as half empty.
A realist drinks it no matter how much there is.