Yes, the 14" Studebaker rims are 1958 only, and even then you could option 15" rims like other years. I have seen exactly 2 Studebakers on the road outside of International Meets with 14" rims, a 58 Silver Hawk and a 58 Commander 4 dr sedan. Rare...
That being said, the 15" Studebaker rims are easy to find. Don't bother to search out for (63 disk) 64-66 all Studebaker rims, as any should be fine for the much smaller and lighter Henry J. If you are driving a HJ with non factory power as in V8, then I would suggest looking for the Ford LTD/Mercury Marquis 15 inch rims up to 1992 (when they went to 16 inch), and especially if you are using radials. Cheap, and since they fit Studebakers, then I suppose they should fit Henry Js as well. IF you use Ford rims, you MUST use the Ford lug nuts, as the Studebaker lugs are fine for Studebakers, but TOO small for the lug holes of the Ford rims to be safe.
Just thought of this--would the HJ caps have clips like the Studebakers did? I take you mean caps and not wheel covers which are a different beast. The modern rims are a bit different and might not be suitable for either). I do know that my 50's Studebaker rims have the slots for hub caps even though all my 50's Studebakers came with Deluxe full size wheel covers.
The trick is likely not the rims, but finding radials that are tall and thin enough to be a good fit. Modern radials are too short and wide. I suppose if you can find something like a 205/75/15 that would be close, but a 215/75/15 might be better if you could find one. Back when 195 size tires were available, they were much closer to the width of an HJ tire, which was pretty skinny. Do a circumference test of an OEM bias tire on the rim if you have one, and see how much less the circumference is with a 205/75 or 215/75. Do the math to see how close, smaller or larger it is than OEM percentage-wise and you can figure out what sort of percentage your speedo and engine revs will change.