Methinks you will find that a TS (terminal services) approach will be the way to go. That means your db and either a runtime or full Access version would be both hosted remotely. Your terminal simply transmits user input to the remote host. The downside is that there's no way you'd get to keep using the old version. The upsides would be that you don't continue to expose yourself to the vulnerabilities of outdated software, plus you would not be trying to transmit data over the internet. There's no way I would attempt to use an Access db over a WAN, which is what you'd be doing over host suppliers such as OneDrive, Google drive, etc.
The more we hear silence, the more we begin to think about our value in this universe.
Paraphrase of Professor Brian Cox.