In general, possibly not at all.

Indeed, without loss of generality $a_i\ne0$ for all $i$. Then the pdf of each $a_iU_i$ is log concave and hence (by the well-known Proposition 3.5) the pdf of $S_N$ is log concave.

So, if the pdf of $X$ is substantially not log concave (say U-shaped), then $X$ cannot be approximated by $S_N$ in distribution.

In general, possibly not at all.

Indeed, without loss of generality $a_i\ne0$ for all $i$. Then the pdf of each $a_iU_i$ is log concave and hence (by the well-known Proposition 3.5) the pdf of $S_N$ is log concave.

So, if the pdf of $X$ is substantially not log concave (say U-shaped), then $X$ cannot be approximated by $S_N$ in distribution, even with a however large $N$.