Richard Firsten opines, "actually, there is nothing ungrammatical with the question Should not they have permission . . . ?, but it's certainly very uncommon and probably never heard or even written anymore. A very long time ago this construction was sometimes used."
It appears then that in Firsten's reckoning, anything that was once grammatical in English is always grammatical in English. There are probably some who believe him, but to them, I say, "Buton ğam anum, ğæt ğæt Leden and ğæt Englisc nabbağ na ane wisan on ğære spræce fandunge."
Personally, I'd say it sounds ungrammatical. There are no instances of the construction in the BNC and but a single example in the COCA:
why should not they manage their own schools, Love asked?But if shouldn't is simply a contraction of should not, as Firsten says it is, then what's the problem. The problem is that shouldn't is NOT simply a contraction, but an actual negative form. Most auxiliary verbs have negative forms: isn't, wasn't, won't, mustn't, haven't, didn't, etc. But if this were a mere contraction, it wouldn't be limited to auxiliary verbs. Contracted is, for example, can but stuck onto anything at all (in speech at least):
- my name's Brett (noun)
- his's better than yours (possessive pronoun)
- the place we went to's not there anymore (preposition)
- the bag that's too big's lying there (adjective)
- You can not go to church and still be a good Christian.
- You can't go to church and still be a good Christian.