Based on your question, you have developed an application which is going to be shared by multiple systems(I think it's going to use the single database, like entries from all machines goes to single database).(1 server for database & many other machines for application)
If the above statement is true then, you don't have to install database on every machine. Install the database only in server & install the application on other machines.
Add Prerequisites of .NET Framework in Visual Studio Setup Project[
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It that statement is false then I think you have developed an application only for single machine which includes the database itself. And you're going to install the application for different clients.( (1 machine with database & application) * No of clients)
Deploy your Application and Database[
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Create database during installation of a .NET application - Version 2[
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