Does Bounded Covergence Theorem hold for Riemann integral?

No. Enumerate the rationals in [0,1] with the sequence $\{r_n\}_{n=1}^\infty$. Now define $f_n(x)$ by $f_n(x) = 1$ if $x = r_k$ for some $1\le k \le n$ and 0 otherwise. For all $n$, we have $$\int_0^1 f_n(x)\,dx = 0.$$ However, the limit function, the indicator of the rationals in $[0,1]$ is not Riemann integrable. The bounded convergence theorem fails for the Riemann Integral.

A dominated convergence theorem for the Riemann integral exists, due to Arzel`a. But one needs the addtional assumption that the limit function is Riemann integrable, since this does not follow from pointwise bounded convergence. For a proof see either W. A. J. Luxemburg: Arzela's Dominated Convergence Theorem for the Riemann Integral. The American Mathematical Monthly, Vol. 78, No. 9 (Nov., 1971), 970-979 or the book "An interactive introduction to mathematical analysis" by J. Lewin, Cambridge Univ. Press, 2003, 2014.

But this statement is true:

Let $\{f_n\}$ be a sequence of Riemann Integrable functions such that $f_n:[a,b]\rightarrow\mathbb{R}$ and $|f_n(x)|<M$ for all $n\geq1$ with $M>0$. Suppose that $f_n\to f$ pointwise where $f:[a,b]\rightarrow\mathbb{R}$ is Riemann Integrable. Then $$\lim\limits_{n\to\infty}\int_a^bf_n(x)\,dx = \int_a^bf(x)\,dx$$