To answer your question, first of all you need to clearly mention which kind of service (Web Service or Windows Service)
And I would like to mention here that there is no simple converters available to do it.
In case of Web Service:
You need to copy your Business Logic from the WPF application and create a new web service of your choice. You can keep your Data Layer also.
In case of Windows Service
Your wpf application can run under Service Controller as a windows server, you can find a workaround here
Answers for the other questions you have asked
1. How this can be transformed into a program for the client?
Obviously you need to have an installer project in case of Windows Service, if it is a web service then you need to deploy it on IIS either Local or Global.
2. Which database platform do they might use to storage the data generated by the system?
First of all you need to get this information from your client. Its better to stick with SQL Server since you have already developed it.
3. There's a database program for the client's side?
It is also based on your database engine which you have already developed with. If you use localdb then it will be very easy to get it installed in your client machine. For other types of database engines you may need to have a separate database engine installation (you may need an licence based on the version)
4. When the program is converted, an installer (.exe) is generated?
You can create your own installer project or you can take advantage of .Net Deployment Techniques, here you can see more information about it.
Update
5. How to install SQL Server at client machine?
From your comments, i understood the actual question is different, this is the answer you are looking for. So there is a detailed stackoverflow question here, take a look.