There are such versions of the law of the iterated logarithm even for independent random vectors in an arbitrary separable Banach space. See e.g. Theorems 4.1 and 4.2.
In the case when the Banach space is a Hilbert one, Theorem 4.1 implies the following:
Theorem 1: Let $X,X_1,X_2,\dots$ be iid random vectors in a separable Hilbert space $(H,\langle\cdot,\cdot\rangle,|\cdot|)$ with $EX=0$ and $E|X|^2<\infty$. Let $S_n:=X_1+\cdots+X_n$. Then $$\limsup_n\frac{|S_n|}{\sqrt{2n\ln\ln n}}=\sigma$$ almost surely, where $$\sigma:=\sup\big\{\sqrt{E\langle X,f\rangle^2}\colon f\in H,|f|=1|\big\}.$$
To deduce Theorem 1 from Theorem 4.1, one only needs to note the following two points:
(i) In view of formula (2.5), $\sup_{x\in K}|x|=\sigma$.
(ii) $E|S_n|\le\sqrt{E|S_n|^2}=\sqrt{nE|X|^2}=o(\sqrt{2n\ln\ln n})$, so that condition (ii) of Theorem 4.1 holds.
In the case when $H=\mathbb R$, Theorem 1 becomes the law of the iterated logarithm cited in your post.