Can knowledge exist without structure?
The answer to this question is no, and it relates to the definitions of knowledge and structure.
Knowledge is often taken by epistemologists to be some sort of verified belief, and the process of verification whether it be establishment of adequate justification and truth or otherwise (thanks to the Gettier problem) subscribes to the idea that some form of inference is required to move the state of a proposition from belief to knowledge status. It is on the definition of inference, then that the answer to your question hinges.
What is inference? For some people, justification may exclusively be based on intuition such as those who believe in divine revelation. But most philosophers, including many theologians, reject this, and instead rely on deduction, induction, and abduction to justify conclusions from premises. A quick survey of history of philosophy will reveal that this is a central metaphysical aim.
Ultimately, then, if you, like the Ancient Greek physiki, reject revelation soley, and instead rely on reason, then you are relying on structure for your knowledge. How? Well deduction is a structural pattern among premises and conclusions. In modus ponens, for instance, P then Q and P imply Q regardless of any P and Q. So:
P1 If Socrates is in the kitchen, he is in the house.
P2 Socrates is in the kitchen.
C Socrates is in the house.
The certainty (which is the aim of characterizing belief as knowledge) is established by the structure of the method of justification, in this case deduction. A weaker form of certainty can be established with induction, though Hume noted its problems.
P1 Socrates is often in the kitchen on Mondays.
P2 Today is Monday.
C It is likely Socrates is in the kitchen.
Notice how language an frequency and modality make this an entirely different argument. Note that logicians consider deduction a far more reliable method of justification than induction.
So, does knowledge rely on structure? Yes, if one takes the simple introductory of definitions of knowledge and structure as used by philosophers generally, and reduces them to more fundamental meanings in ordinary language, one can conclude with certainty that logical structure is necessary for knowledge.
This is a fact to many of analytical philosophers know to be true. If you're interested in the connection between objectivity and knowledge, you might want to start with the logical positivists like Mach and Hempel form the Vienna and Berlin circles and move your way forward to the present day.