I would put them in a terrarium (a ten gallon fish tank is excellent and cheap too), cover with plastic or vinyl (a Jiffy seed lid works fairly well), and put them under lights to put some size on before introducing them to their ultimate growing spot. Water as needed without making the media sloppy wet. Watch out for excessive condensation and vent if needed. On the flip side watch out for drying and water if you see wilt and the soil is dry. Hint: often if the soil is wet and if the plants are wilted then more than likely the roots are rotting - if this is the case, pull the plant out of the media ASAP and cut the rot out which might mean cutting the entire root ball out and more, then repot to a more porous media.

I've lost many over the years from the VB (and other vendors) due to treating them as "ready to go" plants. When it finally dawned on me that these are baby plants and are probably grown in perfect conditions then I changed my methodology and have had almost 100% success with growing baby plants on.

When they fill their original pots then move them to the next size pot and start weaning them from the terrarium. If they wilt then cover them again and continue to introduce them to more "normal" air until they can be removed from the terrarium.

As for replanting them, I would stay with a tiny pot with drainage holes, fill with good potting mix, add water, put a planting hole in the middle, and carefully transfer the tiny plant with whatever roots you can salvage to the new pot. Backfill the planting hole to cover the rootball. I often use Solo 2 oz rinse cups with holes cut on the bottom rim filled with perlite to start new begonias which is comparable to what the VB does (they use a lot of vermiculite instead of perlite).

Remember that the VB sells most of their plants as terrarium plants but often some of the begonias can become very large plants.

Here is one of mine from the VB when I let it grow to whatever it wanted to: This is Shamus.
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This is Maid Marion in a very large cookie jar:
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Here are some rexes I got from them but they quickly filled this tank and when I removed them I lost most of them. Moral of the story is to slowly introduce them to the outside air.
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Finally here is Queen Olympus which took a few years to get her to the outside of a tank but she is doing just fine.

Here she is inside a round fish bowl with other begonias last year before moving it outdoors.

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And here she is after spending summer outdoors:
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hcmcdole, those are beautiful! Thanks for your advice. I will keep my eye open for used tank. Sheesh, just gave away a few big and small ones . . . . what is wrong with me?

thank you!
Sherry