Distribution of standard Ito integral is well known: $$I_1(f) = \int_0^T f(t)dB(t) \sim \mathcal{N}\bigg(0, \int_0^T f^2(t)dt\bigg).$$ Is it possible to find the distribution of multiple WienerIto integral: $$I_n(f) = \int_{[0,T]^n} f(t_1, \dots, t_n) dB(t_1) \cdots dB(t_n)$$

1$\begingroup$ I don't know of a closed form expression, but there are certainly lots of results about such distributions. You might like to look up "Wiener chaos". $\endgroup$ – Nate Eldredge Aug 26 '18 at 19:22
Below some references regarding distributional properties of Wiener chaoses
The book, Gaussian Hilbert spaces, by S. Janson, is a standard reference to start with. In particular, you might want to read Chapters 2, 5 and 6. Chapter 6 contains general tail bound.
For precise moment and tails estimates, you can have a look at this: https://projecteuclid.org/euclid.aop/1171377444.
Regarding absolute continuity of the laws, you might want to check these: https://projecteuclid.org/euclid.kjm/1250522278 and https://projecteuclid.org/euclid.pja/1195517581
Regarding anticoncentration, Theorem 8 of http://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0008/0003/a001/ can help.
Finally, there are many open questions when the order of the multiple WienerItô integral is greater than 3.