There are no player facing methods for these things.
Epic Boons and Supernatural Gifts are DM facing options to be used as quest rewards. As such, there are no player facing choices that guarantee receipt of one of these gifts. Supernatural Gifts are described:
A supernatural gift is a special reward granted by a being or force of great magical power. Such supernatural gifts come in two forms: blessings and charms. A blessing is usually bestowed by a god or a godlike being. A charm is typically the work of a powerful spirit, a location of ancient magic, or a creature that has legendary actions.
These things are given as options to the Dungeon Master for rewarding players for completing quests or other tasks incident to epic adventuring. We see this often in published adventures – these supernatural gifts, charms, and blessings are given as quest rewards at some point in most adventures.
Epic Boons are generally reserved for the DM to give rewards to player characters who have reached 20th level, but as you have observed, the Aberrant Dragonmark feat gives the DM an option for providing an Epic Boon to character as early as 10th level:
At the DM’s option, a character who has the Aberrant Dragonmark feat has a chance of manifesting greater power. Upon reaching 10th level, such a character has a 10 percent chance of gaining an epic boon from among the options in chapter 7 of the Dungeon Master’s Guide. If the character fails to gain a boon, they have a 10 percent chance the next time they gain a level.
I have bolded the crucial phrase here: at the DM’s option. Even taking the Aberrant Dragonmark feat does not make gaining an Epic Boon a player facing decision – it is still entirely up to the DM to grant the boon, even after a player has chosen the feat.
Characters created for a Theros Campaign start with a Theros-specific gift.
We see similar features given in Mythic Odysseys of Theros, but these features differ from the ones in question in two distinct ways:
- The list of gifts is unique to the Theros setting
- Every character gets one to start the adventure
So this is sort of what you are looking for, but the application is quite specific to one setting and does not reference the gifts and boons from Chapter 7 of the DMG at all. It is also worth noting that the gifts here are generally much stronger than the blessings and charms, akin more to full feats.