Let $LG=\operatorname{Maps}(S^1,G)$ be the loop group of a compact Lie group $G$. I should add some adjectives to $G$, but for sake of simplicity let's just take $G=SU(2)$.

There is a central extension $$1\to S^1\to\widetilde{LG}\to LG\to 1$$ (these are classified by "level" in $H^3(G)$, but we may as well restrict attention to the "universal" such extension corresponding to a generator of this group). The constructions of this central extension that I have found so far (e.g. the one in Pressley--Segal) all go via first defining a closed $2$-form on $LG$, arguing it defines a unique $S^1$-bundle, and then putting a group structure on this bundle.

Is there a more intrinsic definition of $\widetilde{LG}$?

In other words, given a loop $\gamma:S^1\to G$, I would like to have an intrinsically defined principal $S^1$ homogeneous space (or, equivalently, a $1$-dimensional complex vector space).

For example, here is an answer "up to homotopy". Since $\pi_1(G)=\pi_2(G)=0$ and $\pi_3(G)=\mathbb Z$, given any loop $\gamma:S^1\to G$, the space of extensions $\bar\gamma:D^2\to G$ (i.e. $\bar\gamma|_{\partial D^2=S^1}=\gamma$) is homotopy equivalent to $\Omega^2G$ which is connected with fundamental group $\mathbb Z$. If we take the $1$-truncation of this space (add cells to kill all higher homotopy groups), we get $S^1$ (up to homotopy).

This gives an "intrinsically defined" space homotopy equivalent to $S^1$ defined in terms of a given loop $\gamma:S^1\to G$ (although it's not very explicit, and has questionable meaning/use). What about an honest one-dimensional complex vector space? (with a natural meaning). Even better, can we define intrinsically a holomorphic line bundle over the complexified loop group $LG_{\mathbb C}$?