You can only acquire Greater Dragonmark in E6 with extreme cheese and a permissive DM.
(tl;dr Dusk Giant cheese is the least questionable method.)
Greater Dragonmark (ECS, p. 54) requires:
Least Dragonmark, Lesser Dragonmark, 12 ranks in any two skills, Member of appropriate dragonmarked race and house
The feat requirements are doable (Lesser Dragonmark requires 9 skill ranks, which you can have at level 6), and the fluff requirement is fine as long as you're playing an Eberron campaign. It's the 12 skill ranks that are going to cause problems.
Normally, your maximum skill ranks are your character level + 3, meaning you would cap out at 9 skill ranks as a level 6 character. There are a few ways to get around this, but they're very cheesy and most of them won't be available in an E6 campaign.
Note that I did find one E6 variant that lists Skilled Beyond Your Years as giving +3 to maximum skill ranks, rather than +2. However, since you can't normally take the same feat twice, that will still only get you one of your two skills to 12 ranks.
Beyond that, you'll need to look into cheesier methods. Let's go through them one by one.
The Primary Contact feat (Cityscape, p. 61) lets you gain one bonus skill rank, even if doing so would put you above your maximum ranks. So far, so good. Unfortunately, it includes the following restriction:
This feat cannot be taken more than once. If the primary contact associated with this feat dies or is otherwise removed from the campaign, the DM can, at his discretion, either replace that contact with a new contact from the same organization or allow you to name one of your other contacts as your primary contact. In either event, you do not gain the bonus skill rank a second time, but neither do you lose it just because your contact has left the campaign.
This feat will let you get a single skill up to 10 ranks (or 12 with Skilled Beyond Your Years), but it won't help you get two skills up to 12 ranks each.
Bonus HD + Psychic Reformation
By gaining temporary bonus hit dice, you can raise your character level, which in turn raises your maximum skill ranks. Then, you can use the psionic power Psychic Reformation to move skill points into the required skills, and if your DM is feeling friendly they might allow you to keep skills above cap even after the temporary hit dice expire.
You're going to run into problems doing this in an E6 campaign:
- Psychic Reformation is a level 4 power. It's not impossible for level 4 powers to exist in an E6 campaign, but it'll need to come from either a monster/NPC with more levels than PCs are allowed to have (i.e., friendly DM), or one of your party members will need to employ some cheese of their own to get access to level 4 powers.
- The standard way of gaining temporary bonus HD is the Bard ability Inspire Greatness. However, Bards gain Inspire Greatness at level 9. Where are you planning to find a level 9 Bard in an E6 campaign?
- An alternate way of gaining HD is to transform yourself into a Barghest while having the feat Assume Supernatural Ability (Savage Species, p. 30), and then use their Feed ability to gain hit dice. This is possible to achieve in E6, but there's another issue: The hit dice gained via the Barghest's Feed ability aren't temporary. That means if you do this, you're permanently becoming a character of higher than 6th level. "You can't be higher than 6th level" is the central tenet of the E6 ruleset; I've never encountered a DM who would allow this in an E6 campaign.
- A better option would be to become a Dusk Giant (Heroes of Horror, p. 147). These creatures have the Cannibalize (Ex) special attack, which lets them eat creatures to gain temporary hit dice. The weakest Dusk Giant has 6 hit dice, making it a valid target for Polymorph for a level 6 character. It doesn't even require Assume Supernatural Ability, since it's (Ex)!
So, if you manage to:
- Find a way to access Polymorph, a level 4 spell, in an E6 campaign
- Turn into a Dusk Giant
- Eat at least 15 people or 60 nonsentient creatures
- Convince your DM to let you be temporarily higher than level 6 in an E6 campaign
- Find a way to access Psychic Reformation, a level 4 power, in an E6 campaign
- Dodge any books you might somehow find flying your way
...then this method will work.
Just be Pun Pun
An inevitable answer to these types of questions is "well you can always just be Pun-Pun" (archive link).
Yep, you sure can. Let us know how your DM feels about that.