I give an explicit counterexample below, so I have modified my answer: No, these sorts of orthogonal coordinates on 'shells' of parallel compact surfaces don't always (or even usually) exist. The reason has to do with whether there are nonconstant functions on the given surface $\mathcal{M}$ that are constant on one of the families of principal curves.
Such functions don't always exist: Suppose you take an oriented surface $\mathcal{M}\subset\mathbb{E}^3$, maybe, say, an embedded torus such that it has no umbilics and such that at least one family of its principal curves (i.e., the lines of curvature) wind densely around the torus. It then turns out that, for $|t|<\epsilon$, the oriented-distance $t$ parallel surface $\mathcal{M}_t$ (where $\mathcal{M} = \mathcal{M}_0$) also has these properties. An example is constructed below. [Note that the standard ellipsoid is very different: The lines of curvature on an ellipsoid are all closed. However, this is very unusual behavior. Most differential geometry books give the ellipsoid example because it's pretty and computable, and this tends to make students think that lines of curvature are 'usually' closed, but they are not.]
I claim that, for a surface as above, the desired functions $f_2$ and $f_3$ will not exist on the $\epsilon$-shell $\mathcal{S}$ around $\mathcal{M}$. The reason is that each noncritical level set of $f_2=c$ (for example) will be closed in $\mathcal{S}$ and will have to intersect each $\mathcal{M}_t$ in a line of curvature, which, by supposition, is a dense curve in $\mathcal{M}_t$. Thus, this (closed) level set will have to include $\mathcal{M}_t$. Hence, this level set must be a disjoint union of the parallel surfaces $\mathcal{M}_t$. It follows that $\nabla f_1$ and $\nabla f_2$ must be parallel along any noncritical level set of $f_2$. However, this is absurd, because $\nabla f_1$ and $\nabla f_2$ must be perpendicular wherever $\nabla f_2$ is nonvanishing and $\nabla f_2$ does not vanish along a noncritical level set (by definition). It follows, then that $f_2$ cannot have any noncritical level sets, i.e., $\nabla f_2$ must vanish identically. Since you don't consider this to be a nontrivial solution, there are no nontrivial solutions.
Added comment (an explicit example): It turns out that there's an easy explicit example of a torus for which there is no nontrivial solution. Let $C\subset\mathbb{E}^3$ be a closed, embedded, regular space curve, parametrized by arclength $ds$. Thus, associated to the inclusion mapping $X:C\to\mathbb{E}^3$, there will be its Frenet apparatus, i.e., an orthonormal frame field $T,N,B:C\to S^2$ and functions $\kappa,\tau:C\to\mathbb{R}$ with $\kappa >0$ such that
$$
dX = T\ ds,\quad dT = \kappa N\ ds,\quad dN = (\tau B -\kappa T)\ ds,\quad
dB = -\tau N\ ds.
$$
It is easy to see that there are such curves with $\int_C \tau\ ds$ being an irrational multiple of $\pi$, so let's assume this. Now, for sufficiently small $t>0$, consider the tube of radius $t$ about $C$, parametrized by $Y_t:C\times S^1\to\mathbb{E}^3$ where
$$
Y_t = X + t\cos\theta\ N + t\sin\theta\ B.
$$
(We require only that $t$ be so small that $Y_t$ be a smooth embedding of $C\times S^1$. In particular, we want $t$ to be less than $1/\kappa_{max}$.) This is a parallel family of surfaces.
It is easy to compute that the principal curves on $Y_t$ are the leaves of the two foliations $ds=0$ and $d\theta + \tau(s)\ ds = 0$. Because I required that $\int_C\tau\ ds$ be an irrational multiple of $\pi$, it follows that each of the leaves of $d\theta + \tau(s)\ ds = 0$ is dense in $C\times S^1$. Consequently, the above argument applies to show that, even though you can take $f_1$ to be $t$ for this family, and, say $f_3$ to be some function of $s$, any function $f_2$ on a shell constructed by taking $t$ in some small interval in $(0,1/\kappa_{max})$ that satisfies $d f_2 \wedge(d\theta + \tau(s)\ ds) =0$ will have to be constant on the level sets of $f_1$.
One further remark: It turns out that a necessary and sufficient condition for the existence of a non-trival solution $(f_2,f_3)$ for the given $f_1$ is simply this: A nontrivial (in the OP's sense) solution exists if and only if there exist functions $g_2$ and $g_3$ on $\mathcal{M}$ with the properties that $dg_2\wedge dg_3$ is nonvanishing on a dense open set in $\mathcal{M}$ and that each $g_i$ is constant on a family of principal curves on $\mathcal{M}$. Sufficiency follows since one can simply take $f_i = \pi^*(g_i)$ for $i=2$ or $3$ where $\pi:\mathcal{S}\to\mathcal{M}$ is the retraction of the shell $\mathcal{S}$ to $\mathcal{M}$ long the normal lines from $\mathcal{M}$. Necessity follows since, if $f_2$ and $f_3$ exist, then one can simply define $g_i$ to be the restriction of $f_i$ to $\mathcal{M}$. Thus, one can see that the behavior of the principal curves on $\mathcal{M}$ completely determines whether or not there are solutions.
Here, I try to give a exemple for each of the number one and three curves with a convex polyhedron where I just draw the net. The geodesic are the dots lines (which are staight on the net). I think that one can also construct the second one as well by changing the angles of the triangle in the first exemple.
I don't know any theorem that gives a general answer for your question. My only gess is that if you don't have a trivial estimate that says that on a subset $B$, $\int_B K \geq 4\pi $, then we should be able to construct a exemple.
EDIT : I just give some details of what anton says in his comment : in each small loop the integral of curvature is $\pi+\alpha_i$ with $\alpha_i$ the angle of the crossing. Because the curvature is positive the sum of the angles of the triangle is larger than $\pi$ and then the total curvature is larger than $3\pi+ \sum \alpha_i > 4\pi$. (However with a polyhedron we can still obtain an equality)
Best Answer
As pointed out in the comment by Daniel Asimov, the osculating sphere is undefined (or may not exist).
But, one may ask about the size of the maximal ball surrounded by $\Sigma$ in terms of its principal curvatures --- this is the closest relative of your 1-dimensional question.
This question was considered in a sequence of papers by Vladimir Lagunov and Abram Fet; they constructed examples of surfaces with principal curvatures at most 1 that do not surround a ball of radius 1. You may check 11C in our book What is differential geometry... + you may check an this video by Sergio Zamora. Plus, here is artwork Ana Cristina Cháliz Cáliz from our book that illustrates one such example; this is the so-called Lagunov's fishbowl.