The best place to repair a 1950s refrigerator is a shop that specializes in refurbishing and selling antique appliances. Many of these, such as AntiqueAppliances.com, only work with older machines and have a significant inventory of parts that can be otherwise difficult to source.
Some shops repair both modern and vintage appliances, such as Complete Appliance Repair Utah. These businesses market themselves as being able to source parts and complete repairs for appliances from multiple eras, but getting references from other vintage customers is wise to keep from receiving substandard service. When restored, a 1950s refrigerator is a key element in creating a retro look in the kitchen.