The distribution of $X$ is the distribution of the ratio $$\frac{\sum_{i=1}^n\lambda_iZ_i^2}{\sum_{i=1}^nZ_i^2} $$ of two quadratic forms in iid standard normal random variables $Z_1,\dots,Z_n$. The distribution of this ratio was studied by Gurland; also see e.g. Watson.

The distribution of $X$ is the distribution of the ratio $$\frac{\sum_{i=1}^n\lambda_iZ_i^2}{\sum_{i=1}^nZ_i^2} $$ of two quadratic forms in iid standard normal random variables $Z_1,\dots,Z_n$ (because the distribution of $(v_1,\dots,v_n)$ is the same as that of $(Z_1,\dots,Z_n)\big/\sqrt{\sum_{i=1}^nZ_i^2}$). The distribution of such ratios was studied by Gurland; also see e.g. Watson.