Is there a method to obtain Mammout Coin without buy them?

In short, no.

The only way to obtain Mammoth coins is to spend your IRL dollars on them.

If you don't feel like paying for just the coins, most DLC packs come with a set amount of Mammoth coins including their exclusive contents. These packs are also giftable, so if you have a generous friend, there's always a chance they decide to give you it as a gift. The coins are still being bought, but at least you are getting them for free ;)

Edit: I also wouldn't put it past the developers to offer mammoth coins as a Viewership Reward. This is currently not the case, but who knows.