While it's not a record card, I came across some pictures of the aircraft log book for the TBM-3E that's in the Intrepid Sea, Air & Space Museum the other day:

(Source:
Aerial Visuals)

(Source:
Aerial Visuals)

(Source:
Aerial Visuals)

(Source:
Aerial Visuals)
Furthermore, it's interesting to note that in addition to the aircraft log book above, both the Tri-State Warbird Museum and the EAA Aviation Museum have the original U.S. Navy aircraft log books for their respective FG-1Ds.
[1] (The aircraft history card mentioned in a
previous post in another thread was found within the pages of the the former.)
Note that there appears to be a second log book off to the side in the top picture. This is particularly interesting as the TSWM's example is actually made up of three separate logs books stuck together to make a single volume.
Lastly, being able to see another log book is also useful in that it confirms that U.S. Navy aircraft log books from the World War II era were bound in a booklet with a hard, brown cover. (Indeed, apparently all U.S. Navy log books of the period were made in such a way, as examples (
1,
2,
3,
4,
5) of U.S. Navy "Aviator's Flight Log Books", N.AER. 4111, from the
vintage aviation logbooks thread have a matching style.) So for anyone searching through a collection for a log book or requesting a copy from an a museum should be able to easily either visually identify it or describe what to look for.