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][1], $Z_1^2+Z_2^2+Z_3^2 \sim \chi^2_3$.

The square root of a $\chi^2_3$ distribution  is a [chi distribution with three degrees of freedom][1]. 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](https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution).


  [1]: https://en.wikipedia.org/wiki/Chi-squared_distribution#Definition