I'm a financial application developer with big dependence on databases.
You need to figure out how many users and connections to the database they require. I'm an MS SQL Server (Express or even LocalDb for cheap option) fan because of the nice tooling and integration with Visual Studio. Really the easiest option if you're a .NET developer and already in the Microsoft ecosystem. If they stay small, they will manage with the free versions, but you need to judge the possible growth in number of users and at some stage they might need a proper licence in order to accommodate the number of simultaneous connections. Without know too much, I'm guessing that they'll manage with the free versions for quite a while.
If you want to use open-source / always free tools, MySQL is an options. This has been forked to MariaDB and most of the ongoing development seems to be on MariaDB. From your perspective MariaDB will be fully MySQL compliant and you can almost view them as interchangeable. The fork is pretty much because of the same reason as when OpenOffice forked to LibreOffice when Oracle bought OpenOffice.
Also open-source/free is PostgreSQL. Personally I prefer this to MySQL.
You can even look at SQLite, which is kind of like the open-source alternative to Microsoft Access (technically the JET database) in that it's a database in a single file that you can copy around. It doesn't come with its own front-end like Access though. It's stable and fast and easy to set up. Even if you don't end up with using this as a final version, it's a good tool to start your development with.
There are also the so-called NoSQL family of databases like MongoDB, which breaks away from the usual paradigm of having static schema tables. It's a different mindset and I don't have much experience with it. But from what I know is that if you want to easily run reports on the data in your database, it's not as easy.
If you're developing your front-end on Windows, you can find ODBC drivers for all of these above.
For the front-end, I would recommend staying away from VBA. It's just really limited and old. Download Visual Studio 2019 Community Edition (free) and start playing with VB.NET (very similar syntax to VBA) or C#.
Also look at using an Object Relational Mapper (ORM) like Dapper or EntityFramework. Bit of a learning curve there, but it will make your code database-agnostic, so in the end you don't have to worry about which database you use (with some caveats).