Is it possible to prove in the set theory NBG (with local choice but without global choice) that the proper class of ordinals injects in every proper class ?
The answer is no. That principle is equivalent to global choice.
To see this, consider the class $W$ consisting of all well-orderings of any rank-initial segment $V_\alpha$, for any $\alpha$. If we had an injection of Ord into $W$, then there must be unboundedly many $\alpha$'s that are used, since each $V_\alpha$ has only a set-sized family of well-orderings. Thus, we have a global selection of well-orderings of unboundedly many $V_\alpha$, and from this we can define a well-ordering of the entire universe. Namely, $x<y$ if the rank of $x$ is lower than that of $y$, or if they have the same rank and $x<y$ in the first well-ordering of some sufficiently large $V_\alpha$ to appear in the range of the injection of Ord into $W$.
Update. I made a blog post concerning these various equivalent formulations of The global choice principle in Gödel-Bernays set theory, in which I explain this answer and give several other related formulations and arguments.