The following phrases:
Looked at me funny
Looking at me funny
Don’t sound grammatically correct, but I hear them as turns of phrase relatively frequently. It should be something like:
Looked at me funnily
Looking at me funnily
But then the original sentiment of the phrase seems lost, as it’s so seldom used. Is it valid for the adjective "funny" to implicitly modify the noun "the look" that is being received, or is this just another case of standardised bad grammar?
Yes, it is correct, though it is arguably restricted to informal speech and writing.
Funny is here an adverb, with the meaning ‘in an odd or peculiar way’. This usage is recorded e.g. in the Merriam-Webster dictionary (you’ll need to scroll down to the meaning of the word as an adverb). It is also recorded in the OED, with the note regional and colloq.