Do pilots ever suffer PTSD after emergency landings?
Yes, for a while: Capt. Al Haynes, after saving 184 lives on United Flight 232.

I was lucky enough to have dinner with Al Haynes a few years ago, and he told me that he had a very difficult time overcoming the guilt of the disaster. The talks he gives have been a way of coping.  (As a side note, he also donates all the money he receives from the talks to the survivors of the crash).
Capt. Haynes has said the same thing in various interviews:
The Seattle Times:

Haynes said it took many sessions with a psychiatrist to overcome his
survivor’s guilt. He resumed flying three months after the accident;
mandatory retirement at 60 followed two years later. [link]

NY Mag:

It was very hard to get past the guilt of surviving. My job had been
to get people from point A to point B safely, and I didn’t do it. I
felt that I had killed them. I had a lot of psychiatric help at the
hospital and afterward. Finally, one psychiatrist said, “You’re not
going to find an answer. You just have to accept that it happened.” [link]

AvWeb:

Haynes said, at the time, that he remained troubled by the accident
and constantly wondered if there was something he could have done,
some overlooked decision that would have saved all aboard. [link]

Before his death, Denny Fitch said something similar:

&quot;To find out 112 people didn't make it, that just about destroyed me,&quot;
Fitch once said. &quot;I would have given my life for any of them. It was a
really tough time.&quot; [link]