There are no player facing methods for these things, with one possible exception.
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 level1, 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.
User F1000003 observes in their answer that it is possible to use true polymorph to transform into a chwinga, which can then dispense 1 supernatural charm per day. Your mileage may vary here - the presence of chwingas in your setting is going to be up to the DM, and your DM may rule that you can only polymorph into a creature you have seen before. This method also requires a 17th level spellcaster with true polymorph or access to the spell through a scroll. So it's there, but only accessible in pretty narrow circumstances that are still largely dependent on the DM.
Theros and Ravenloft have setting specific gifts and mechanics.
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.2
Additionally, in Van Richten's Guide to Ravenloft, we see another character creation and development option - Dark Gifts:
A character can select a Dark Gift from the “Dark Gifts Descriptions” section. This supernatural gift expresses both a mysterious power and insidious influence. Work with your DM to determine how your character gained this Dark Gift. Is it the manifestation of a family curse? Is it a reward for a sacrifice you made at a forgotten shrine? Did you bargain with a voice whispering from a mirror, the sea, or the Mists? Does the Dark Gift compound with your other character choices to reinforce your unique origin? Each Dark Gift can be expressed in various ways, with the following options exploring various manifestations to spark your imagination.
Dark Gifts are intended for starting characters, but characters who don’t choose one might be presented with opportunities to gain a Dark Gift as their stories—and desperate circumstances—unfold.
The available gifts are again specific to the setting, but they do provide a player facing option for gaining them, but are again not the ones found in Chapter 7 of the DMG.
1 DMG, Chapter 7, pg. 231, Epic Boons: "An epic boon is a special power available only to 20th level characters. Characters at that level gain such boons only if you want them to and only when you feel it’s appropriate."
2 As an alternative to the gifts unique to Mythic Odysseys of Theros, the section on gifts also offers several Player's Handbook feats as alternatives: "If your campaign uses the optional feat rules from chapter 6 of the Player’s Handbook, your Dungeon Master might allow you to take a feat as a variant supernatural gift."