What version of Sage 100 are you using?

100 Advanced ERP 2015.

Displaying the PDF after it has been printed only occurs when using Paperless Office for printing Journals & Registers.

The Paperless Office Delivery Options in Customer Maintenance will allow for setting up multiple recipients of electronic delivery of documents (in your example Sales Orders). For each customer you can choose to deliver to the email address on the Sales Order (which can be changed in Sales Order Entry for each entry), the customer's email address from the Main tab in Customer Maintenance, a selection of Contact Codes for the customer each with an email address, a selection of specific other email addresses.

I'm trying to follow.
When you do Sales Order Entry and quick print? Or print the whole batch of SO's with the status of 'New'?
Why do you want them to go to the server rather than the Paperless Viewer?
When you set up paperless, you can designate that SO's get emailed to:
one or more contacts from Customer Maintenance
The email addresss tied to the sales order
or a pre-defined set of email addresses for that customer.
Have you talked to your Sage partner about this stuff?

We were on an old WindX system that automatically popped up an email with the pdf attachment right there. All the sales rep had to do was put in the email address and hit send.
My boss won't allow anyone to see the stored pdfs nor get into the feature where all you have to do is select the pdf and click email. Also, only one email address is setup and they're not allowing them to put in multiple address, though the reps want that.
And we were told by our Partner to store Paperless Office documents on the server--we have a shared folder on a server for such a purpose but were told to store them on the Sage server.
I tried showing them the Export feature to generate a pdf on their system but "OH MAN, I have to find it, then click on it and then attach it??? That's too much work!" They want things the way they were with the old fast, flat-file system where the programmer had it all set up for them. I was just wondering if there was a vb or ms script that can be written to generate this pdf when they use "pdf all" so we get them stored as my boss wants and they can also email them from the generated pdf. Thank you for your responses, everyone.

Set the paperless options for the customer to use the transaction's email address, then the user enters the address to use there and everything happens automatically, without a pop-up. If they want multiples, without being allowed to set that up, then set the transaction's the email address to themselves, and they can forward the received email to whoever they wish.
Paperless Office is designed to not use the workstation's email client. If you want to script something where you let Paperless create the PDF, then find the right record in the table to get the file name / path, and create a new email with the attachment... well... good luck with that. Especially if you print in batches.