To form an exciton you need at least two conditions:
- an electron and hole.
- overlap of wavefunction (proximity).
In bulk materials you also need low temperatures, however in semiconductor nanostructures it is possible to have excitons at room temperature because of the quantum confinement of both carrier types (in a type-I quantum well or quantum dot for example).
As Ruslan pointed out above, if you photo-excite into an electron accepting state in the bandgap you have satisfied the first criteria, however not the second because the hole the photogenerated hole remains is the valence band. Traps can accept either electrons or holes depending on the type of defect or impurity, so although you pin one carrier type but the carrier with opposite charge will always remain free.
It is possible to get defects/impurities which accept both carriers types, but these are not trap centres, they are recombination centres. For example, if the site pins a hole it may also attract an electron resulting in rapid recombination. We don't normally think of these sites forming excitions because, presumably, the lifetime of the state is extremely short. However, in principle I don't see why it isn't possible, but it might just be an insignificant effect.
Maybe some more insight could be gained by consider the quantum mechanics of exitonic wavefunction vs. the wavefunction for recombination centres.