So I'll admit it's taken a long time to research this question, but my conclusion is thus:
They were teenagers in love, unaware that they couldn't just elope.
My believe is that they could not have directly eloped, and would have to be on the run for an additional two years to satisfy Japanese law.
Not just that, but the reality of figuring out how to survive while living like that - a subject touched on towards the end of the series, but not really fully fleshed out - would've likely proven too hard for them to survive. Without any steady source of income, and even with a place to stay, their chances of surviving without a larger support network (yes, besides their friends) would've been very, very small.
However, there are loopholes to this scenario. That is, there are reasonable scenarios which could have occurred that would've cleared these hurdles.
Please note: these are conjectural, and based only on the evidence we have at hand. I wouldn't say at all that this would occur; I wanted to explore the possibility.
Starting with Taiga, we can see that there are cracks in her relationships with her parents. At this point, we know this because
Taiga initially stormed off because her mother is pregnant, and will be introducing a new child into her family. Taiga takes umbrage to this, and would rather not be in her mother's family. Her father has already left town and would not be in a position to be a "parent" of any kind to her.
Given that, and how their relationship had progressed at that point, it could be seen that Taiga could have received approval from her mother to get married.
Note that in the last episode, a voice mail left by Taiga's mom which sounded like furious resignation could've cleared the path to her becoming emancipated (as I as an American understand it), and would've been the start of her being able to get married before 20.
From Ryuuji's side, his perspective is a lot less clear due to his familial circumstances, but there is an angle that could have been exploited.
After he ran off with Taiga, Yasuko had started to see Ryuuji as more of an independent man. Even though her burning wish was to see her son succeed academically - a part in which she failed and genuinely wanted her son not to fail - after his brief but sharp rebuke, she could be seen as respecting Ryuuji just a bit more.
Even though Yasuko was heartbroken, it could be conjectured that Ryuuji could've asked her for her approval...after she was found.
Note that Yasuko cut off all communication with the world, essentially, after Ryuuji ran off. The only way that anyone got in touch with her was by telling her that Ryuuji was with her parents.
That would've opened up the dialogue somewhat, and could have allowed Ryuuji to gain his mother's permission to marry.