Are you referring to the motor/frame part of an electric stand mixer? If so, you'll need to know the brand & then perhaps you can find the model by googling. I'd start the search on ebay & other places that sell antique & kitchen collectibles, looking for 'antique mixer'; I'm not sure if they were always called 'stand' mixers.

I'm not sure what you mean by "mixing bowl stand". Were the stand and bowl used with a hand-held mixer or for mixing by hand, or do they belong to a stand mixer? Whatever the case, you'll need to know the name of the manufacturer. Is the bowl clear or milk glass? What size is it? Does it have any markings on the bottom? For example, if it's labelled "Pyrex", it would have been made by Corning. If the bowl is distinctive, maybe someone (not me!) could ID the manufacturer if you post a photo here. It probably isn't as daunting as it sounds -- in the 40's and 50's, most items would have been made in the U.S. There are lots of sites devoted to vintage kitchenware, so chances are good that you'll find some information at least.

I second Antiquesilver's suggestion to search on eBay. Unfortunately, many of the sellers don't identify or describe items accurately, so I recommend setting up a few different searches using different terminology. I would also set up a separate search to find a matching bowl. If you find one, you might be able to determine the manufacturer and get a better idea of its age.

Note that you can set up a search so that eBay notifies you when there's a match, but if you're not careful about it, you'll be flooded with false notifications. Vintage mixing bowls are very common, so you'll need to put some thought into how you search. If you make the searches too generic, you won't have time to wade through all the results. If they're too specific, you may miss out on a match because the seller didn't provide accurate or sufficient information. Also, be aware that a lot of sellers aren't very good at spelling, so include "alternate" spellings in your search.

Here is a link that might be useful: Pyrex Pattern Identification Site