On the expression "no [noun 1] or any [noun 2]"

Solution 1:

First of all, you can analyze the sentence as

I have no (allergies or any medical issues)

and your uneasiness about the negation can be resolved. The Corpus of Contemporary American English (COCA) has a few examples of people using this. However, you could also use "nor" instead of "or":

I have no allergies, nor any medical issues.

The COCA has examples of that as well. Some people are uneasy with the use of "nor" without "neither" so for full prescriptive correctness you could say

John serves a chicken with neither sauce nor any kind of seasoning.

which is indisputably standard and grammatical, but in casual speech it sounds a bit stuffy and many normal humans don't speak that way.

Solution 2:

Arguably the more "technically correct" form would be...

I have no allergies nor any [other] medical issues.

...but in practice OP's examples are quite common.

Note that in both these examples the word "other" is at least implicit (even if not actually present), in that both "denied possibilities" (allergies|medical issues, and sauce|kind of seasoning) consist of a specific example, followed by the general classification for "other things of that same type".

The word "any" simply conveys optional emphasis (definitely nothing of the type referred to applies). I feel this strengthens the case for using nor rather than or, because nor also emphasises the negatory nature of the statement.

It's a fine point, but looking at a similar construction without an implicit "other"...

I have no ability or|nor [any] desire to answer this question...

...it seems to me that in the absence of any other context, we might more often expect or to be used if the statement continued with something like "...but since I'm legally obliged to, I will answer". Whereas the more explicitly negative nor might be followed by "...and therefore refuse to answer".