Question: I would like to create a system in which anyone can hook up their email, and have all emails that come to them go through my server first. Anyone who signs up for this service, will have their email vetted by my servers filtration system. If it passes through the filters, I'd like to forward that email onto their inbox.

Answer: 
The vetting part will be quite simple. The difficult part will be ensuring that the user's email passes through your server first. This will either require the users installing some sort of software on their system to intercept the mail coming from their mailboxes, or it will require the users changing their email to a domain you host (maybe a subdomain of their primary domain delegated to you?) or it will require the admins of the mail systems forwarding mail to you when it arrives for your subscribed users.

None of these things are easy, simple to implement, or foolproof.
This is perfectly possible. There are plenty of services out there that make a business out of doing it. The difficult bit is not in configuring their email servers to send email via you, or configuring your servers to accept, and then forward their email but instead in producing useful filters that can't be done "locally" to the sending servers and therefore making it worth your and their effort.

You're effectively acting as a smarthost for the users - instead of publishing MX records to the net and sending email from their own servers, they publish your server(s) as the MX record for their incoming email and forward all their outgoing email to you from their edge SMTP server.

It's not exactly a difficult configuration on any reasonable email server.