Take another look at the sentence:
Lying under the dining room table, Noodle the poodle waited for bites of
roast beef that Rosie sneaked off her plate.
Lying under the dining room table, a participle phrase, comes right before the noun Noodle. This is a clear and logical pairing of target and modifier. There is no error.
Before you continue, review the rules for misplaced and dangling modifiers.