This is just for fun! The title pretty much says it all. It's probably a very difficult question.
Up to the $40,000^{th}$ prime $(479909)$, I have found only $5$, $71$ and $369119$ with this property. Somebody with better hardware than me might have better luck!
Edit: Sivaram Ambikasaran has checked that these are the only ones up to $10^8$, i.e. up to the $5761455^{th}$ prime (see the comments).
(Here is a very naive heuristic: if we suppose the sum $S_n$ of primes less than $p_n$ to be randomly distributed mod $p_n$, it will be divisible by $p_n$ with probability $1/p_n$. Hence the function $f$ given by
$f(n) = \begin{cases} 1, & \text{ if } p_n \mid S_n, \\ 0, &\text{ otherwise.} \end{cases}$
should have expected value $1/p_n$, and hence I'd expect the series $\displaystyle\sum_{n\geq 1}f(n)$ to diverge very slowly, like the sum of reciprocals of the primes, which is approximately $\log \log n$... but then again, such an argument is more or less worthless.)
Cheers!