This will be simple and quick. No, I doubt it would have PTSD and the usual statements that are suppose to back it up isn't useful such as these statements from this one's top comment when googling as it is apart of a topped reddit post:
-Jotaro loses his shit when Kira says muda
-He is hesitant to/dislikes using time stop. Doesn't use it all unless in combat (part PTSD, part the fact that time stop puts a heavy burden on Jotaro's internal organs, mainly his heart. It is why he can't push past 5 seconds, but DIO can. Stated by Araki)
-Is visibly shaken when talking about DIO/the Agents of DIO in part 6
-Loses his shit when Pucci throws knives at Jolyne in the same manner DIO did to him
Also, you can just infer from a lot of his mannerisms that he holds everything inside and is deeply sad/suffering when remembering the crusaders (his brief cameo in Part 5). Not all PTSD manifests as panic.
Now some of the comment did debunk this users claims but this post is from me, personally, now if you watch the show than some of this wouldn't make sense nor indicate some sort of PTSD.
A strong reason why Jotaro acts a bit differently after Part 3 is literally from his aging and maturing. Infact...
17 (Part 3)
28 (Part 4)
30 (Part 5)
40s (Part 6)
Jotaro was still a teenager in Part 3 happen and when Part 4 begun, it was 11 years after part 3.
I would say, some of the claims that are called out can be seen as misinterpretations such as him "[Losing] his shit when Pucci throws knives at Jolyne in the same manner DIO did to him" because Jolyne is in danger and unlike when Jotaro dealt with this back in Part 3. He saw what DIO was doing in that moment during the time stop and was able to counter it (for the most part) unlike Jolyne where she didn't know what Pucci was doing during the Time Stop.
Jotaro cares about Jolyne, she's her father so him panicking isn't a sign of PTSD, it is a worry about her dying.
Now why would someone have this fan theory in there is head, beats me. I feel like it comes from the appeal of it since it feels like it gives Jotaro depth even though it literally doesn't make sense with the claims that are suppose to support it.