The only thing I'm aware of in Sage 100 that requires a stoppage in real world activity is the inventory Physical Count function (to ensure accurate counts).  In ancient versions (meaning 12+yrs old) period end was more delicate a process, but modern versions handle that very differently. 

We have customers who never run Period End, and go months in between reconciliations.  It sounds like your business processes are the limiting factor, not the software.

Perhaps reach out to your Sage Partner / Consultant for a process review.

I wondered if that was actually the case. It seemed insane to me that you can't finalize an invoice for up to two weeks a month. Thereby leaving you blind to what has shipped and what hasn't, the open order report completely inaccurate, what your inventory really looks like for allocations and on hand. Not to mention the inability to provide an accurate account statement to anyone or even have a snap shot of what your sales look like for the month.

You can absolutely continue to post invoices / shipments before the accounting month end activities have been completed.  During the posting process, just set an appropriate posting date.

You may want to reach out to your Reseller of Record for Best Practices related to this inquiry.  It sounds like your accounting/finance department is following ancient processes/procedures; that even then had proper procedures to continue forward into the new month.  

What version of Sage are you using? 

As Kevin stated the modern versions allow you to keep going without having to close the previous period (though I do like to close the previous period as soon as possible).  Does sound like this is a business process or possibly a training issue.

1. Check with the person who told you that it wasn't possible in order to uncover their reasoning

Everyone suspects this is an internal issue and not a limitation of Sage 100.  I suspect "accounting" might need to talk with your reseller on best practices

The version being used is the current version upgraded 2 months ago.

I would agree it's both a training problem and a business process.