Yes, Amazing Charts will continue to work on your own on-prem server if you stop paying support. However, some functionality will stop but nothing of consequence. However, there are some things to consider:
As time moves on, your server will become more and more out-of-date. Eventually, you will be on an unsupported version of Amazing Charts with an unsupported version of Windows and an unsupported version of SQL Server. You will also be on an unsupported backup software. This situation will happen because you aren't paying for AC support so no new version of AC which means no new version of SQL Server which means no new version of Windows.
Should your server fail at that point, assuming the data is recoverable at all, you will have little help in getting AC back up and running again from AC support since it will likely have been YEARS since you've been a client and are on an old version of AC. They'll help you get up on the latest version but that means paying them lots of money and buying a new server.
Keep in mind your mileage may vary. The situation I'm laying out is very possible but also leans towards the extreme. However, it is a situation I've seen more than once and don't like to be involved in.
If you are moving to another system, I can convert your AC data and imported items to one PDF per patient. You can either keep those PDF files on a computer for reference or possibly import them into your new EMR as imported items (or whatever they call it). PM me for details. If you decide to just let your server run for the number of years you will need it, completely unsupported, you are rolling the dice. If you decide to take that path, go with God.
JamesNT