The system shows real-time availability, but performs an exact search on the dates entered. Some chalets have different arrival and departure days (e.g. Sunday-Sunday, Saturday-Friday) or different minimum stays, so they may not appear in results even if available on nearby dates.
If you have flexible dates, try entering only the arrival date and the following day, or dates close to your preferred ones: you will then see more properties and can check the full availability calendar for each chalet.