According to Wikipedia, the number of permutations in $S_n$ with exactly $k$ inversions is the coefficient of $X^k$ in $$1(1+X)(1+X+X^2)\cdots(1+X+\cdots+X^{n-1}).$$ Denote this by $c(n,k)$. This shows that $$c(n+1,k) = \sum_{l=0}^k c(n,k-l).$$ So the number of permutations in $S_n$ with at most $k$ inversions is equal to the number of permutations in $S_{n+1}$ with exactly $k$ inversions. This has a neat combinatorial proof as well (hint: take $\pi\in S_{n+1}$ and remove $n+1$).
If we are interested only in the coefficient of $X^k$, then factors $X^m$ for $m > k$ don't make any difference. So for $n > k$, $c(n,k)$ is the coefficient of $X^k$ in $$ \begin{align*} &1(1+X)\cdots(1+X+\cdots+X^{k-1}) (1+X+\cdots+X^k+\cdots)^{n-k} \\ = &1(1+X)\cdots(1+X+\cdots+X^{k-1}) \frac{1}{(1-X)^{n-k}} \\ = &1(1+X)\cdots(1+X+\cdots+X^{k-1}) \sum_{t=0}^\infty \binom{t+n-k-1}{t} X^t. \end{align*} $$ This implies the formula $$ c(n,k) = \sum_{t=0}^k \binom{n+t-k-1}{t} c(k,k-t), \quad n > k. $$
When $k$ is constant, the asymptotically most important term is the one corresponding to $t = k$, and we have $$ c(n,k) = \binom{n-1}{k} + O_k(n^{k-1}) = \frac{1}{k!} n^k + O_k(n^{k-1}). $$ The same asymptotics work for $c(n+1,k)$, which is what you were after.
For non-constant $k$, using the fact that $\binom{n+t-k-1}{t} = \binom{n+t-k-1}{n-k-1}$ is increasing in $t$ and $\sum_{t=0}^k c(k,t) \leq k!$, we get the bounds $$ \binom{n-1}{k} \leq c(n,k) \leq k! \binom{n-1}{k}. $$ Better bounds are surely possible, but I'll leave that to you.