If I have an issue with the SQL Server, and need to roll back to a
prior version of the database, will the File Recovery option from the
VM backup be adequate?
Maybe. VM Backups don't always give you consistent SQL backups. They usually work, but not always. If you have everything setup just right and get consistent VM backups, it might be ok-- but you are running a fairly old OS on that VM, so I'd be nervous. Very nervous. If the data is really important to you, then you should backup the data, not just the VM. Sometimes you want to restore just the data to another VM to investigate, not the entire server. I also hope you have more than just "last night's VM backup" at any given time. Sometimes bad things happen on Friday and you don't notice until Monday.
Or should I also be running SQL backups via a maintenance plan on the
server on the VM?
Yes, you should be running SQL backups if you your data is important. If your data is really important (you don't want to lose half a day of it), you should be doing full backups periodically (e.g. nightly) transaction log backups many times per hour, and keeping a few weeks worth of backups in rotation. If your data is super-important (you don't want to lose more than a few seconds), you should be mirroring it over to another database server in near-time (asynchronously). If it is critical (you don't want to lose any data), then you want to mirror to another server in real-time (synchronously).
Of course, if you are already running in Azure and don't have a DBA, managing a database is a lot easier, safer, more available, and generally cheaper if you use Azure SQL rather than trying to manage your own instance SQL Server in a VM-- oh yeah, and backups are handled for you, with millisecond point-in-time recovery for up to 45 days-- and they handle the mirroring for you to. If you want to mirror to another region across the country, you do have to pay extra for that, though.
