Hi Rik586,
There are some resources introducing the detailed information about working with an Access database on SharePoint. The following things are for your reference. You can follow up the instructions in the articles to narrow down the root cause of the issue firstly.
Help article: Move your data to Access and Office 365
Help article: Build and publish an Access database to SharePoint
A similar thread: Customer Database
Training sites: Creating Web Databases with Access 2010 and Access Services
If the issue persists, it will be nice if you can provide more details about Publish Failed, such as screenshots?
Thanks,
Maggie Li