My knowledge of this subject is obsolete, but anyway here are some partial answers.

If you have a Ricci curvature bound in addition to the injectivity radius, then you can recover the diffeomorphism type, e.g. with harmonic coordinates, see M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 102 (1990), no. 2, 429-445. But 100 should be replaced by some radius depending on the dimension and the Ricci curvature bound.

**Correction**. It is the constant 1 that would depend on the Ricci curvature bound. (I did the rescaling argument wrong.)

Without curvature bounds things get more complicated. It is very easy to recover *homotopy* type, In fact, you don't need injectivity radius, only a contractibility function: every ball of radius $r\le r_0$ is contractible within a ball of radius $f(r)$. Given this, you can construct homotopy equivalence from an almost isometry defined on a net: extend it step by step to skeletons of a cell decomposition, just make sure that $n$ iterations of $f$ (or maybe $2f$) do not grow bigger than your constant 100.

If the dimension is not 3 (and maybe for 3 as well, since the Poincare conjecture is now solved), you can recover *homeomorphism* type under a pre-compactness assumption (in your set-up, this pre-compactness boils down to something like that every ball of radius 2 is covered by a bounded number of balls of radius 1, and then your radius 100 depends on this number). This follows from the arguments in Grove-Petersen-Wu, Geometric finiteness theorems via controlled topology. Invent. Math. 99 (1990), no. 1, 205-213.

**Correction.** Again, the pre-compactness would translate into something more complicated after rescaling the injectivity radius to 1.

They also use only local contractibility function, and in this more general context diffeomorphism stability fails in dimension 4. Of course this does not answer your question since the injectivity radius assumption is so much stronger. I don't know how essential the pre-compactness is. They certainly need it for their result (which is finiteness of topology types) but perhaps it may be relaxed if you only need stability.