Are there any Army records, maybe Company records, which list and name the enlisted men in each platoon? In other words, are there any records which could tell me which platoon my wwii relative served in?
If you know the company he served in, you could start with the company morning reports. Any other source will most likely be relying on those morning reports to establish who was with the company exactly when and where. They will show when a man was killed, wounded, or otherwise transferred to or from the company. Morning reports from WW2 are housed in St. Louis with the National Archives. You can read more about accessing those records here: Access to Morning Reports and Unit Rosters
I don't know of any official records available that have platoon level rosters. I have (and still am) researching several family members who served in WWII and only got that level of information on one. In that case, I happened to get in contact with someone who had interviewed several veterans of my uncle's Company. Between us, we were able to figure out that my uncle was in the same foxhole as one of the men he interviewed. To say that I "happened" to get in contact with him suggests it was luck. To be sure, there was an element of serendipity involved. I was doing some volunteer research for my uncle's regiment association and mentioned that I had an uncle in "G Company". The next thing I knew, I was referred to the "Company G expert" and the rest is history. I guess what I'm trying to say is that sometimes you find what you're looking for when you're not looking for it, if that makes any sense.
I suspect that platoon records would be difficult to come by. The membership changed according to losses, transfers, etc.Company MRs might give you those changes but it would be catch-as-catch-can. If you get lucky like Tommy you might come across someone who can fill you in, but it would be the product of diligent research.
Morning Reports are a great resource, but I have yet to see a Company MR that indicated platoon assignments. I'm not saying it can't happen, but it would certainly be outside the norm if it did.