local on premise single database for desktop as well as web application

I'm involved in a new project where my client is keen on having a local on premise database for their daily business transactions.The requirement is to build a desktop (windows based) application which will connect to the local database server. However in future they also want their customers to be able to view information contained in that local database. for example: view their profile,transactions etc.

Our first approach, was in suggesting them to opt for a web application altogether.But since web application would be dependent on the network speed and considering the critically of everyday data entry/updates, they are insisting on having a desktop application for their inhouse operations.

Is it possible to build a desktop LAN based app and then later build a web application which the user can interact that ultimately talks to a single local database ? or in other words windows desktop app and a web app connecting to a single local database...