Can a long polymer chain interact with itself via van der Waals forces?
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.