Is 'might be able to' grammatically correct?

There is no grammatical restriction on using the adjective able with modal verbs. We can use the adjective able with the verbs may, might, can and could, for instance. Here are some examples sentences using BE + the adjective able:

• People can be able to change but unwilling to change.
• She wished she could be able to say yes if he asked her again
• I may be able to help you with that.
• They might not be able to come.

While it's often true that using some modal verbs with able will cause some uncomfortable redundancy, this isn't always the case. Modal verbs can often indicate other things apart from ability. So if, for example, we are using one of these modal verbs in its epistemic sense, in other words as a modal of deduction, then we can more or less freely use it with able.

There is an interesting article which quotes the examples in Shakespeare, and explains why things have changed since then and:

As the modal system developed, the association of can with be able to fell away, for two reasons. There was an overlap of meaning between the ability sense of can and that of be able to, which made "can be able to" seem tautologous. And "be able to" came to fill the gap of expressing ability whenever a nonfinite construction required it: one says "to be able to talk", not to can talk. The two expressions thus complemented each other, and this made it less likely that they would co-occur.

And that such usage survives in some dialects and non-native speakers:

But the usage didn't die away completely, especially in negative expressions. I have heard can't be able to in some regional dialects. Nor did it entirely disappear from Standard English. If you do a Google search for 'cannot be able to' there are 3000 or so examples. Now, admittedly many of these are likely to come from people for whom English is a second or foreign language, and where learning levels are unclear, but there are several cases where the text has been written by native speakers in Standard English.

From 2008 things have changed and a search in google books gives some 50,000 results.

Checking with a dictionary you find that:

may (and can #8) have different meanings, the first one listed by Oxford is:

1. used to say that something is possible

   • That may or may not be true.
   • He may have (= perhaps he has) missed his train.
   • They may well win.
   • There is a range of programs on the market which may be described as design aids.
2. used when admitting that something is true before introducing another point, argument, etc.
   • He may be a good father but he's a terrible husband.

In your example:

"They might not be able to ask me anything, however...."

the verb might has this particular meaning, and the sentence is correct because it doesn't mean 'to be able to', it is not tautologous since it is equivalent to:

"It is quite possible they are not able to ask me anythyng,..."

You ask: "Is 'might be able to' grammatically correct?", rules of grammar are not involved here, is is a matter of logic, semantics and style: tautologies should be avoided