"I know a man with a wooden leg named Smith"

The 1964 Walt Disney film Mary Poppins features the following famous lines:

Bert: I know a man with a wooden leg named Smith.
Uncle Albert: What's the name of his other leg?

It is a joke that exploits a common ambiguity in English communication.

Obviously, naming a wooden leg is absurd, and so most listeners understand that the intention of Bert's reply, upon hearing it, is to explain that the name belongs to the man. This discrepancy makes Albert's reply into the punchline.

Assuming, for argument, that naming a wooden leg were not absurd, is either interpretation, of naming the man versus naming the leg, more fully supported by English grammar?

No, not really, although it could maybe depend on what you mean by "more fully supported by English grammar".

Neither interpretation is ungrammatical at all in the model of English grammar that is standardly accepted by syntacticians. It's very clear that English grammar rules produce ambiguous output in many situations. Grammar rules are not normally conceived of as providing degrees of support for one structure vs. another: rather, they either allow or forbid a certain structure. (There may be optional grammar rules, or rules that only exist for certain speakers, but that's different from rules that provide a gradient amount of support for constructions.)

You could only speak of grammar providing greater or lesser amounts of support if you don't view "grammar" as consisting of rules about which grammatical structures are allowed. I don't really know what it means to do this, but I think some people seem to have opinions like "grammar is really a sociological phenomenon" or "grammar is really just about communication". If you have that kind of viewpoint, however, I think it would be hard to exclude context and expectations from your definition of "grammar", so I wouldn't say it makes much sense to speak of what grammar says about the sentence "I know a man with a wooden leg named Smith". It would say different things depending on the context.

Xanne in the comments mentions a supposed tendency in certain cases to go with "the nearest reference" but although that's a somewhat common idea that English speakers have about usage in some areas of language, it's not a rule about what is allowed, nor is it clear that it's really an accurate description of a case like this: the two possible options for what "named Smith" can be attached to are "a wooden leg" and "a man with a wooden leg". Both come right before "named Smith"; "a wooden leg" is not actually closer. The difference is in bracketing.