Which one of these is right?
Is there a way [...]?
Is there any way [...]?
May I use either? If yes, what's the difference?
The expression a way alludes to one particular way. The expression any way alludes to one of any number of possible ways.
As you mentioned in your astute comment, at the time of asking the question, there's really no difference. Yes, you can use either one of them – in this context.
I would say that these questions have no difference in meaning. Why? Let's assume I know three ways to accomplish this task. No matter which question you asked me, my answer would be the same:
I would not be more or less inclined to mention only one method if you used a way instead of any way, or vice-versa.
In other contexts, though, the words a and any are not interchangeable. In fact, in some contexts, replacing one with the other would result in an ungrammatical statement:
But, in the context of asking a question about solving a problem, either one is just fine:
I have to partially
it is often the case that the two words are interchangeable. However, this is only the case when there's a neutral tone to the question. For instance, the question
works just as well as
But if the question has an implicit extra meaning, then this is no longer the case. For instance, after spending several hours trying to draw a perfect circle, one might yell
In which case
Which is very similar, but has, to my mind, a slightly different tone.
Similarly, if your neighbour is blasting music, one may ask
Which does not work as well with
In summary, I would say that using
Edit: If in doubt, I would favour always using
I think J.R. and Alexis are correct, although I would add that when asking a question in this manner "any" often adds a nuance that the speaker expects that the answer will be "no" or something negative.
"A" sounds neutral, however "B" has a more negative nuance implying that the speaker thinks that answer is no.
Both are correct!