First, make sure the door is depressing the switch - usually a bevelled plastic tab near center at front of reefer section at the top which the door presses on to depress it - or sometimes in the door like the switch on a car door). Also, when you depress the switch you should hear a slight click if it is working right - if not clicking or sticking in depressed state instead of sticking down about 45 degrees of arc then the switch may need replacing - should depress and spring back smoothly. Not tough to replace if you are home handy, otherwise you need an Appliance Repairman - Large. Of course, if electrical handy you can use a volt-ohm meter to check the bulb socket for power when depressing the door switch - if getting power then definitely bulbor short base that is the problem, if no power then you have to progress to dismantling cover to see if switch or socket is the problem.
If that seems OK, check the new bulb in another light socket (assuming it is a standard base size bulb or you have a light like a chandelier or desk lamp or lava lamp that accepts its base type) to be sure it is not defective.
Then, if you can find another bulb with same base size (generally standard Medium Edison base - normal light bulb base) that will fit in the space in the reefer, try that. If it works, then the problem is with the new bulb - and since you already tested the bulb elsewhere to prove it works, the only thing left is that the base is not contacting the spring contact in the socket, as Tim said.
Unplug reefer and let sit 15 minutes to let any capacitor mostly discharge, then carefully bridge the screwdriver metal across the two flat prongs on the cord (remember - it is already unplugged from the wall here) to be sure any starter capacitor is discharged. Do NOT do this without the 15 minute delay - you can zap the capacitor.
Then carefully, with a flat tip screwdriveror needle nose pliers (and probably another person holding a flashlight and a mirror too unless you remove all the the shelves to get up inside) reach down inside the socket and gently pry the center spring contact up till it sticks up a bit more so the bulb does not have to go in as far to contact it. Many of the new machine-made bulbs (I have especially noticed this with GE's) now have almost no solder "button" on the bottom of the bulb, so the older bulb with a pronounced "button" may have pressed the center prong down to where a new bulb with almost no solder button does not make contact with it.