The vast majority of accommodation in Svalbard is owned by companies and reserved for their employees. The few private apartments that are there tend make more money as short term holiday rentals and on sites like AirBnB so very few become available for 'normal' long term rental. The easiest way to get a place to live is to get a job there that comes with an apartment.

That being said there are a fair few private houses on the island and there is almost always something for sale. Here you can see what's currently available: https://www.finn.no/realestate/homes/search.html?q=longyearb...