As I understand it, proteins interact with each other and other structures primarily through natural vibrations that arise from their folded structure. Are there any non-organic molecules that are so large that some parts of them could act as binding sites or receptors such as in proteins?
Proteins are strings of amino acids, there are 20 common amino acids, they all have a common backbone NH2-CHR-COOH, where for each amino acid the R group is different, called the side chain. These amino acids string together to form the protein NH2-CHR1-CO-NH-CHR2-CO-...-NH-CHRn-COOH, n can be anywhere from just a few to millions. Each protein therefore is a distinct set of amino acids in a specific pattern.
The different amino acid side chain groups interact with other chemicals in specific ways, and you can imagine a protein is like a long string which gets folded into a specific knot with a very specific shape. So there are two things going on when the protein interacts with other molecules: the shape and the side chains. Another thing that happens is if the protein interacts with another molecule it often changes the shape of the protein, so then the protein will react differently to other molecules before or after binding to that molecule.
Some of the properties of proteins are shared by RNA, some scientists have suggested that RNA carried out the functions of proteins in early life before proteins were developed.
Now, back to your question: There are two things that allow proteins to do what they do, their size and their variability (the alphabet of 20 different amino acids). So to find an equivalent class of chemicals we would not only need something very large, but also something which can form distinct patterns. Large is not a problem, there are plenty of classes of inorganic molecules which form networks that make proteins look tiny, but these are generally made by simply repeating the same unit over and over again. Basically, the amazing thing about carbon is its ability to form long polymer chains with side groups. The closest thing I can think of would be some sort of silicone, which has an inorganic backbone of -Si-O-Si-O-, but silicones are still organic because they have carbons hanging off the silicon backbone.
There are examples of inorganic materials which have specific sized binding sites, they are used to filter out materials (like how molecular sieves adsorb water to make anhydrous liquids) or as catalysts (like zeolites, for an example see
http://en.wikipedia.org/wiki/ZSM-5). So I guess the answer to your question is yes, there are inorganic materials with structurally specific binding spots.