TL; DR:
This is a case of the relative pronoun que with omitted antecedent ("sillas", or "luces"), which makes it stressed (qué).
This seems to be a modification of the more standard phrase "(no) tener qué + verb". For example,
No tenía qué leer
which means "He did not have anything to read". This example is given in Section of the Nueva Gramática de la Lengua Española (NGLE) as a "uso tónico del relativo qué". In Section 22.1c, such uses are documented:
Pueden ser tónicos o átonos los relativos que incorporan tácitamente antecedentes, como los que se construyen en relativas libres con los verbos tener y haber: [...] No tengo quien me ayude.
i.e., que is a relative pronoun, and when its antecedent is omitted, it can be stressed (have an accent). This means that
- No tenía (nada/ningún libro) que leer
becomes
by omitting the antecedent "nada" or "ningún libro" and adding an accent to que.
In Section 26.12i, an example of the use of "qué" as a relative pronoun with omitted antecedent (relativo sin antecedente expreso) and introduced by a preposition is given:
una artista como yo no tiene con qué comprar (Vargas, Pasado)
This example is very similar to
- Los asistentes no tienen en qué sentarse
which is formed by omitting the antecedent "sillas" in the sentence "Los asistentes no tienen sillas en (las) que sentarse", and adding an accent to que.
In your example, the same thing happens:
- Los asistentes tenían que llevar en qué sentarse
which, again, is formed by omitting "sillas" in "los asistentes tenían que llevar sillas en (las) que sentarse" and adding an accent to que.
Regarding the use of se in alumbrarse, this is a standard reflexive usage: it just means that the attendees had to bring lights to shed light on themselves. Of course, you can argue that what they really want to shed light on is the theater stage, but it is not uncommon to use alumbrarse to mean "shed light to allow oneself to see". The "oneself" there is the reflexive meaning conveyed by se in Spanish.