how do you request a showing?

Your Realtor does this through the MLS. If you are not working with a Realtor, you would have to contact the listing agent directly, either through the info posted on a public site, or via a sign in front of a property. It is a VERY good idea to work with a Realtor who will represent YOUR interests in the transaction!

Get yourself a realtor!!!!!  Many negatives just calling the listing agent-he would then become a dual agent, and even though there are "rules" that govern dual agents,"he" is a  person and people don't always "do the right thing". He gets both commisions. In my opinion, you can't guarentee he'll work diligently on YOUR behalf.  Anything you tell him may be used against you. The same "guy" you want to help you buy your house is also telling the seller what he wants to.  I am sour, but I DID that and lost what I wanted to buy.  Took forever to get out to the house, knew very little info on the house, didn't seem to care, and while I was arranging contractors and inspectors someone else was bidding.  Conversely, I just bought a house and the one I lost is BACK on the market.
You CAN call any realtor to see the house, you might not choose to use that realtor in the future, but for THAT house, you have someone who wants to help you buy a home.