Since you now know what Drill Chuck you have below is a picture that shows the different Drill Chucks supplied with the Shopsmith through the years. Any Drill Chuck with a 5/8" bore will work and that includes non-Shopsmith chucks most of which use an adaptor.
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The above picture is unlabeled since yesterday I confirmed that my previous one was incorrect. Thanks, JPG, for the information.
The Drill Chuck on the bottom left is the original chuck assembly for the Shopsmith Model 10E from 1947. It is a Jacobs 33B Chuck on an adapter. That is followed by the three chucks in the middle of the bottom row the Jacobs 3326 Chuck. The difference on them is the set screw used. The Cushman Chuck on the bottom right replaced the Jacobs Multi-Craft Chucks which are the two on the top row to the left. The Jacobe Multi-Craft Chucks are different only in how the chucks are labeled. The three chucks on the top row from the middle to the right are the different versions of the current Drill Chuck Shopsmith sells. Again, the difference is how they are labeled and the one on the right is the current labeling. These are all 1/2" capacity chucks and use a key. All but the two on the left bottom row use a 5/32" Allen wrench for the set screw to lock them on the Shopsmith's spindle and have come with Mark series tools since the introduction of the Mark 5 in 1954. The other two (bottom row on the left) use a 3/16" Allen wrench and came with the Shopsmith Model 10E or 10ER.
Shopsmith also sells a 5/8" capacity keyless chuck. It can be found at the link below.
https://www.shopsmith.com/ownersite/catalog/keyless_chuck.htm
Something to consider on which Drill Chuck to get is do you plan on using the Shopsmith Hollow Chisel Mortising System with your Shopsmith? The link below is given for reference. If so, it needs one of the regular 1/2" capacity Drill Chucks in the picture above except the original Jacobs 33B Chuck on the adapter. Longer Drill Chuck configurations will not work with the Mortise. That includes many of the non-Shopsmith Drill Chucks I have seen for sale.
https://www.shopsmith.com/ownersite/catalog/rm_hollowchisel.htm