Vierbeins in Schwarzschild

Note that in Schwartschild metric $$ ds^2 = - (1-r_s/r) dt^2 + (1-r_s/r)^{-1} dr^2 + r^2 d Omega_2^2 $$ for $r>r_s$ we have that $t$ is a timelike coordinate and $r$ is a spacelike one; by this I mean that the vectors $partial_t$ and $partial_r$ are, respectively, timelike and spacetlike. In this region of spacetime is therefore meaningful to consider $g_{tt} sim {e_0}^0 = (1-r_s/r)^{1/2}$ and $g_{rr} sim {e_1}^1 = (1-r_s/r)^{-1/2}$.

Then at $r=r_s$ we have the event horizon, and the metric is singular.

Inside the horizon, namely for $r<r_s$, note that $g_{tt}$ and $g_{rr}$ change sign, now $g_{rr}<0$ and $g_{tt}>0$: $t$ is therefore a space coordinate ($partial_t$ is spacelike) and $r$ can be thought as a time coordinate ($partial_r$ is timelike). We'd better, therefore, to relate $g_{rr} sim {e_0}^0 = (r_s/r - 1)^{-1/2}$ and $g_{tt} sim {e_1}^1 = (r_s/r - 1)^{1/2} $.

What we learn from this example is that we had a metric defined in two separate regions of spacetime, and therefore a priori we cannot expect it to have a uniform behaviour. In this case, changing region, we have to change expression to the vierbeins and to change their connection with metric as well.

Note that, having the metric constant signature, the vierbeins always exist: what is to be considered with care is their expression in coordinates.

Some physical intuition is helpful here: recall orthonormal tetrads correspond to timelike observers. In Schwarzschild-Droste coordinates, your definition leads to $$(e_0)^mu=big((1-2M/r)^{-1/2},0,0,0big)$$ Note this is the 4-velocity of a static observer, meaning one with constant $theta$, constant $phi$, and constant $r$. However a static observer can only exist at $r>2M$, so the tetrad (as interpreted this way...) is only meaningful there.

Now for $r<2M$, we can still use your method of rescaling coordinate vectors to construct an orthonormal tetrad, but this time the timelike vector (4-velocity of the corresponding observer) is: $$(e'_0)^mu=Big(0,sqrt{2M/r-1},0,0Big)$$ These observers have $t=textrm{const}$, and may only exist for $r<2M$. They have "energy per mass" $0$.

Yes, you can apply the same procedure for Lemaitre coordinates, and this also yields an orthonormal tetrad, however a different one. In particular, $$(e^textrm{Lemaitre}_0)^mu=(1,0,0,0)$$ in these coordinates, but the same vector has components $$(e^textrm{Lemaitre}_0)^mu=Big((1-2M/r)^{-1},-sqrt{2M/r},0,0Big)$$ in Schwarzschild-Droste coordinates. You may recognise these observers (?) This velocity field can be extended to $r=2M$ (I suggest working in Gullstrand-Painleve coordinates), and thus exists for all $r>0$.