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It is known that van der Waals interaction is an intermolecular forces. So is it possible that a long polymer chain interacting with itself via non-bonded interactions (such as van der Waals forces / electrostatic forces / hydrogen bonding)?

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    $\begingroup$ It can and does. Come to think of it, how can it not to interact? It is just the same atoms governed by the same laws. They don't have a built-in IFF system to tell them whether the other atom is from the same or different molecule. $\endgroup$ – Ivan Neretin Feb 28 at 7:26
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    $\begingroup$ Why these are called intermolecular interactions, they're intramolecular as well, it's just that they're secondary to covalent bonds. $\endgroup$ – Mithoron Feb 28 at 16:48
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You have a possible answer to your question in proteins, an example which includes some long polymer chains.

Intramolecular interactions - while not necessarily the driving force for formation of a collapsed protein globule (usually argued to be due to the hydrophobic effect, requiring intermolecular interactions) - are the basis for higher order structure in proteins and even some forms of RNA. I won't mention DNA since most commonly this forms a bimolecular double helix.

All of the interactions you mention may play a role in stabilizing a particular protein structure or "fold". A good and basic example of protein structure acquisition due to intramolecular interactions in proteins is polyalanine, which acquires alpha-helical secondary structure stabilized by intramolecular hydrogen bonds, typically between amide groups separated by 2 amino acids along the chain. Other types of structure stabilized by intramolecular interactions (H-bonds, usually) in polypeptides include other types of helices, beta sheets and various turns.

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