I would like to confirm if what I did is acceptable.
I am thinking of using the method of moments estimator $\hat{\lambda}_{MM}$ so my challenge here is to evaluate
$$E[X]=\int_0^{\infty}x*\frac{2x}{\lambda^2}e^{-x^2/\lambda^2}dx$$
Using integration by parts, I obtained
$$\therefore \space = \left[ -xe^{-x^2/\lambda^2}+\int e^{-x^2/\lambda^2}dx \right]_{0}^{\infty}$$
I believe that this simplifies to
$$\therefore \space = \lambda \frac{\sqrt{\pi}}{2}$$
So,
$$\hat{\lambda}_{MM} = \frac{2 \bar{X}}{\sqrt{\pi}}$$
is a consistent estimator of $\lambda$.
The asymptotic distribution of this estimator can be obtained from
$$\sqrt{n}(\bar{X}-\lambda \frac{\sqrt{\pi}}{2}) \rightarrow N(0,Var[X])$$
where
$$Var[X] = E[X^2]-E[X]^2 = \lambda^2 (1- \frac{\pi}{4})$$
by CLT so by the delta-method
$$\sqrt{n}(\frac{2 \bar{X}}{\sqrt{\pi}}-\lambda) \rightarrow N(0,\lambda^4(\frac{4}{\pi}-1))$$