Is there a grammar of syntax that takes into account inherent syntactic ambiguity in natural languages?

Yes, cognitive and construction grammars do take ambiguity into account. However, they have to give up a lot of the formal properties of traditional constituency and dependency grammars.

It resolves the particular ambiguities you mention by not having a notion of things like constituents, dependents or complements. It simply treats all surface constructions as units that can be blended together and it is that blending that resolves ambiguities.

So if I say, 'She hit the man with a bottle.' I have certain presuppositional knowledge as well as context to disambiguate (all part of the blend). This means that the context sets up certain mental spaces that will disambiguate this:

• She was bothered by a man in a bar. So she hit the man with a bottle.
• Two men were approaching threateningly each holding a potential blunt instrument. The one holding a plate looked less dangerous. So she hit the man with a bottle.

There is no notion of semantics vs. pragmatics.

Now, this does not solve the problems of something like machine translation where you still have to find a way of 'understanding' the context. But you can build a semantic model, that makes this approach perfectly plausible. However, it won't look very much like the sort of formal grammars you have in mind.

(By the way, constituency and dependency are not 'grammars' per se but rather approaches to syntax. They each cover many 'grammars' that use specific notation and make specific assumptions.)

It is also worth pointing out that conversation analysis has identified a number of patterns in language used to disambiguate - so called conversation repair. Ultimately, context is not always enough. So you may get something like:

• Hold on, she hit the guy holding a bottle, or she took a bottle and hit a guy with it?