Our company has used Access for its database needs. It wants to stick with the current database frontend, but migrate the tables to some cloud based solution. We do not want
Cloud based does not mean you have to think about your database backend any differently, you could it you wished stick with MS access, however as Access does not natively support remote connections then you would need to setup a VPN to your cloud server in order to connect to the .mdb, .aacdb file.
Dedicated database servers are always a nice option (MySql, Sql Server Express, Postgre SQL to mention the free ones) but you will always need to ensure you have the necessary drivers installed (shouldn't be the end of the world)
As for Google forms, I don't really have much experience with those but I imagine Google would have made every effort to ensure they can be implemented relatively easily with existing database products.