Hi,
To create an AWA, you'll need Access Services, which runs on an on-premise SharePoint Server (not on Office 365). Are you saying you have this?
If so, you definitely need to upgrade to Access 2013 because Access 2010 cannot create AWAs. If your company upgrades to SP 2016, the AWA may or may not work depending on any updates or patches MS puts out, which could disable AWAs. However, I would suspect
AWAs won't work with SP 2019 anymore.
Did you know there are other options to AWA? For example, Zoho or Caspio or PowerApps.
Just my 2 cents...