Yes, FacetWP works well in a WordPress multi-site network setup. It can be activated in each sub-site individually, or you can use Network Activation (recommended). But be aware that each sub-site counts against your license’s site limit.

FacetWP will be able to filter content within each individual sub-site. It cannot filter or search across sub-sites, and there a no plans to support this in the future.

See this page for more info about using FacetWP with multi-site.

See also