To become a werewolf, you need to get bit by a relatively rare werewolf in the high level area of your faction's area. They ONLY spawn at night and are usually in pair.
Here are the spawn points for each area:
Daggerfall Covenant, Bangkorai:
Aldmeri Dominion, Reaper's March:
Ebonheart Pact, The Rift:
Apparently there is a moon cycle but a lot of people found werewolf outside of a full moon ( me included ) so I don't think moon matters, including for vampire.
After you're bit (taking damage a few times from it), go to the area's city and a quest marker should show up on your map.
To know if you have the disease or not, check your character menu. The disease is called 'Sanies Lupinus'.
Do the quest and become a werewolf. The quest is actually pretty hard for low level (no heal and solo) so I recommend not trying until you're atleast level 37.
Another way to get it is to let another player bite you by the werewolf shrine but that player has to have the 'bloodmoon' werewolf ability (which let you bite someone once every 7 days). Also don't pay for bite because 99.9% of the time it will be a scam.