Me and other users on the Jacksepticeye Wiki have had issues with a user who repeatedly adding false information to pages and changing images, causing me to revert their edits. Then it became a small edit war where they undid my edits and I undid their edits.
Me and another admin have been blocking their multiple accounts, but they keep coming back. Should I contact FANDOM Staff by email?
Have you tried blocking the user's IP? If so, I would use Special:Contact/user-conflict and explain the situation. Ask them to do an IP check to reasonably verify that these accounts are, in fact, the same person. If they are indeed the same person and they are using different IPs to circumvent the block, you can place, or ask staff to place, a block on that user's entire IP range.
Yes, go ahead and contact them anyways. Based on the block log, it looked like the exact IP should already be blocked. This likely means that (again, assuming this is in fact the same person) a range block will be required. Since you don't know how to do it (nor do I), you can ask staff to do it.