Let $\mathcal{C}(n,m)$ denote the set of sequences you are interested in, namely non-decreasing sequences of length $n$ consisting of integers from $\{0,\ldots,m\}$, and let $C(n,m) = |\mathcal C(n,m)|$ be the number of such sequences. It is not hard to verify the following recursion:
$$
\begin{align*}
&C(0,m) = 1, \\
&C(n,m) = \sum_{k=0}^m C(n-1,k) \text{ for } n > 0.
\end{align*}
$$
In fact, it is not hard to check that
$$ C(n,m) = \binom{n+m}{m}. $$
But the recursion allows us to rank and unrank sequences. More explicitly, we will now describe a bijection $r_{n,m}$ from $\mathcal C(n,m)$ to $[0,C(n,m))$ (i.e., $\{0,\ldots,C(n,m)-1\}$). The bijection is defined recursively as follows (below, $\lambda$ is the empty sequence):
$$
\begin{align*}
&r_{0,m}(\lambda) = 0, \\
&r_{n,m}(\sigma_1,\ldots,\sigma_n) = \sum_{k < \sigma_n} C(n-1,k) + r_{n-1,\sigma_n}(\sigma_1,\ldots,\sigma_{n-1}).
\end{align*}
$$
This doesn't order the sequences in the order you listed, though one can easily adapt it to your order (by considering the first rather than last element). We can also describe the inverse bijection explicitly:
$$
\begin{align*}
&r^{-1}_{0,m}(0) = \lambda, \\
&r^{-1}_{n,m}(r) = r^{-1}_{n-1,k}\left(r - \sum_{\ell<k} C(n-1,\ell)\right),k, \\ &\qquad \text{ where $k$ is defined by } \sum_{\ell < k} C(n-1,\ell) \leq r < \sum_{\ell \leq k} C(n-1,\ell).
\end{align*}
$$
Finally, if all you want is to list the sequences in order, say the order in your post, then you can use the following simple "counter" algorithm. Start with the all-zero sequence $\sigma_1 = \cdots = \sigma_n = 0$. At each step, find the maximal $i$ such that $\sigma_i < m$ (if $\sigma_1 = \cdots = \sigma_n = m$, we are done). Increase $\sigma_i$, and set all numbers following it to its new value. This will enumerate all sequences in lexicographic order, in amortized time $O(1)$ per sequence.
