To the best of my knowledge, you can't update a 2010 web database to 2013 or 2016 (although you can migrate the data by importing the tables into a new custom web app).
An Access web app has limited scope and functionality.
We no longer recommend Access Services for new apps. We will stop creation of new Access-based apps in Share Point Online starting June 2017 and shut down any remaining apps by April 2018.
Without question, the number one feature in the latest release is publishing your database to the web with Share Point.My specialty is Access and Share Point, so what better way to get started than with a series on both?Access and Share Point If you’ve been working with Access for a while, it’s hard to miss the push Microsoft has been putting behind Share Point integration with Access. This is my first post since joining the firm and I look forward to many more!Both Juan and I will be blogging here about Access, SQL Server and Share Point and we love reading your comments, so please leave us feedback below.When we researched how to close these gaps, the answer became clear as well; we’re aligning efforts behind Microsoft Power Apps as the way to build no-code business solutions on desktop and mobile devices.Power Apps offers a comprehensive set of application building tools, connection to custom web APIs, and a wide array of database options including Share Point lists, SQL Azure databases, Common Data Service and third-party data sources.When we first created Access Services in Share Point, we set out on a mission to enable both information workers and developers to quickly create data centric web applications with little or no programming.Over the last several years it has become clear that the needs of our customers have grown beyond the scope of what Access Services can offer, such as mobile device support, integration with line of business data, and professional developer extensions.We have added a feature to export your data to Share Point lists where you can create Power Apps and Microsoft Flows.We have also published guidance on how to port your custom web app to Power Apps here.