Does a one-time pad of digits 0-9 have any weaknesses? I'm working on one of those unsolved puzzles, and one theory I have is that one or more sections of it are using digits of pi, e, or some other irrational number as a one-time pad.
(The obvious approach would just be to try well-known irrational numbers, but I've tried that without success and if it is that there must be some other complication.)
Are there any cryptanalysis methods that that allows for that wouldn't apply to a one-time pad composed of letters?  One thing that occurred to me is that since letters can't be shifted by more than 9 places, it ought to show very small peaks in its frequency distribution, one-tenth the size of ones for plain text and displaced about five letters to the right along the alphabet.  But I don't know how much practical use that would be.
The question is probably equivalent to whether a Gronsfeld cipher has any weaknesses that a Vigenere cipher doesn't.
The one-time pad can be made of digits 0-9, and you can read more about that here. One could also use binary digits, 0 and 1, or letters such as A-Z.  Those three alphabets are the most common.

Are there any cryptanalysis methods that that allows for that wouldn't
apply to a one-time pad composed of letters?

No, there are not.  A properly generated one-time pad is going to have equal strength no matter whether it is composed of letters A-Z, numbers 0-9, or the binary digits 0 and 1.
The CIA, for example, was fond of using one-time pads composed of numbers 0-9 during the Cold War.
However, in practice, using numbers or binary could possibly introduce errors because of fractionation.  In other words, using letters might be more straightforward.

One thing that occurred to me is that since letters can't be shifted
by more than 9 places, it ought to show very small peaks in its
frequency distribution...

Don't worry about that.  It is not the case.

The question is probably equivalent to whether a Gronsfeld cipher has
any weaknesses that a Vigenere cipher doesn't.

One can see what you mean by this, but Gronsfeld and Vigenere are a very different kind of cipher from the one-time pad.