Write $Z_i = \dfrac{X_i}{\sigma} \sim \mathcal{N}(0, 1)$.
Then, $$Y = \sigma\sqrt{Z_1^2+Z_2^2+Z_3^2}\text{.}$$ Using a standard result, $Z_1^2+Z_2^2+Z_3^2 \sim \chi^2_3$.
The square root of a $\chi^2_3$ distribution is a chi distribution with three degrees of freedom. Since we are multiplying this by $\sigma$, we obtain a random variable proportional to a chi distribution with 3 degrees of freedom. Per @whuber's comment, one name for this is the Maxwell-Boltzmann distribution.