Basically you could include the installation of the database server to your application installation. You can for example use a configuration file for the installation and include running a silent installation to your app setup. Have a look at
Install SQL Server 2016 Using a Configuration File[
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However, before doing that few thing you should consider if the database server is always installed when the application is installed.
Typically the database server is installed only once and the same database is used by multiple clients. In this kind of environment including the installation to your application installation would not make sense.
Another thing is that the target site may already have a server installed and they want to use the existing installation, again installing a new server wouldn't be a good thing.
And the third thing is that if the database is installed with the software and it's a single user scenario, then perhaps Express is not the correct edition. Compact is often used in such scenarios and it does not require any separate installation.