The Norwegian (Bokmål) Wikipedia has lists for each county and municipality with all major monuments eligible for the contest. Each list contains the ID number needed when uploading photos for the contest. In addition, the website Kulturminnesøk from the Norwegian Directorate for Cultural Heritage, has all major and minor monuments, including maps for your current location to find what's near. The lists on the Norwegian Bokmål Wikipedia are naturally in Norwegian, but should still be pretty easy to understand no matter what language you speak.
Wikimedia's map tool for Wiki Loves Monuments 2019. This tool doesn't have all eligible monuments, but many. A red label means that a monument still doesn't have any photos on Wikimedia Commons, while a green label means that there are photos. There is no requirement that photos depict monuments that don't yet have photos, so feel free to photograph any monument on the list.
Wikimedia's map tool for Wiki Loves Monuments 2019
Østfold
Akershus
Oslo
Hedmark
Oppland
Buskerud
Vestfold
Telemark
Aust-Agder
Vest-Agder
Rogaland
Hordaland
Sogn og Fjordane
Møre og Romsdal
Trøndelag
Nordland
Troms
Finnmark
Svalbard
