Is there a reason why the game doesn't have a mechanism for trading relics?
The best reason I can think of that this option isn't available is because that would discourage players from solving encounters on the world map, exploring provinces, and thus encountering new neighbors. I mean, sure, you'd get to it
eventually, but the game relying on you obtaining the relics through working on your own incentivizes you to explore the world map.
(Also, you can get relics through Neighborly Help, if you didn't know. It's random, but you can get relics, KP, Power of Provision, MM, and EE spells just by helping your neighboring cities. There are three such random awards available to each player once within the game's preset 24 hour cycle. It resets at 8 PM Eastern, I think.)
Everyone keeps mentioning the tournaments. Without clearing the provinces for that relic ahead of time, though, you aren't going to be able to get the relics you need through the tournament.
That said, I would add to their recommendations of participating in the tournament that you might consider building the Endless Excavation and level it up to get extra relics when you complete a province. Also, keep an eye on the relics slot in the MA crafting window. I ran out of crystal relics in Harandar and had to clear two provinces to get more before any showed up in crafting, so I do feel your pain.