There are a couple of possible variant questions of this, and I'm not quite sure which is appropriate. The first question question is whether, without knowledge of the functor $T$, we could construct $TMF$ by Baas-Sullivan theory.
If you do not invert 6, then this is not possible. Any object constructed by Baas-Sullivan theory has a unit $S \to X$ which factors through $\pi_* S \to \pi_* MU \to \pi_* X$. This violates the fact that the 24-torsion element $\nu \in \pi_3 S$ is detected by any variant of $TMF$.
If you do invert 6, then some variants of this problem have positive solutions and some do not.
The connective spectrum $tmf$ has, away from 6, homotopy groups $\Bbb Z[c_4, c_6, 1/6]$, and can be obtained from $MU$ by inverting 6 and killing a regular sequence.
The version associated to the uncompactified moduli $M_{ell}$, often call $TMF$, has homotopy groups $\Bbb Z[c_4, c_6, \Delta^{-1}, 1/6]$. Because $\Delta^{-1}$ is in negative degrees, this cannot be constructed because any spectrum produced from the Baas-Sullivan method only has positive-degree homotopy groups. This is, however, only a small failure; localization is a relatively mild extra tool to add.
The version associated to the compactified moduli $\overline{M}_{ell}$, sometimes called $Tmf$, has (again, away from 6) the same homotopy groups as $tmf$ in positive degrees and a $\Bbb Z[1/6]$-dual set of groups in negative degrees. The negative homotopy groups obstruct using the Baas-Sullivan method, and it is also not possible to get it simply by localizing something constructed by those techniques. However, it is possible to construct it with a little more work: if we have already constructed $tmf$, then we can get a diagram of localizations
$$
c_4^{-1} tmf \to (c_4 \Delta)^{-1} tmf \leftarrow \Delta^{-1} tmf
$$
whose homotopy pullback is $Tmf$. (This last example illustrates the use of patching together constructions on a cover of $\overline{M}_{ell}$ to get a global construction. It works well because we can construct $tmf$ and then we can do localizations "in the category of $MU$-modules".)
The second question that you might ask is whether we might use Baas-Sullivan theory to construct a diagram (part of $T$) with $TMF$ or $Tmf$ as a homotopy limit. Even if Baas-Sullivan theory were completely functorial, this does not work either: the constructions of Baas-Sullivan theory naturally land in the category of $MU$-modules and so the primary objection about detecting $\nu$ still holds.
A third question that you might ask is whether we might construct objects using Baas-Sullivan theory, and maps between them by some other method, to give us a diagram whose homotopy limit is $[TMF\mid Tmf\mid tmf]$. That's a perfectly good idea (and using Landweber's theorem, after all, is how Morava got this started in "Forms of K-theory").
We run into a couple of problems with this program. One is that Baas-Sullivan theory classically produces objects in the homotopy category, and to construct a homotopy limit we need a lift to an honest diagram. There is no way around this; at some point we have to make sure that it is possible. Another is that, to my knowledge, Baas-Sullivan theory doesn't account for the possibility that we might need to relate the object constructed to another, isomorphic, formal group law constructed by a different regular sequence.
A third is that, as we saw above, to construct such the diagrams we typically want to patch together along open subobjects. That means localization, and localization is generally very difficult to carry out if we only have a multiplication on the homotopy groups rather than on the object proper. Above we were working in $MU$-modules and inverting elements in $MU_*$. Once we leave that situation and go back to situations where the primes 2 and 3 are in play, we no longer have localization available for free. Having a ring in the homotopy category is enough to do a localization if the ring of homotopy groups is commutative, but it's not easy to ensure that the result is still a ring in the homotopy category and allow us to do more localizations after that. Having a diagram of homotopy rings does not ensure a limit object, like a homotopy pullback or a fixed-point object for a group, also has a ring structure. There are also slightly delicate questions about whether localization is unique, or whether it has a universal property of any kind.
These are some reasons why one might move to trying to carry out a construction within a point-set theory of associative or commutative ring spectra. Unfortunately, once we are there Baas-Sullivan theory is no longer in our toolkit; it does not interact perfectly with point-set constructions and products (though Neil Strickland proved a lot about it in "Products on $MU$-modules"). I believe that Laures and McClure have work relating products in bordism theory to spectrum-level products, but I'm unsure of the degree to which one can make it run with singularity bordism.
