As far as I know it is indeed theoretically possible to catch chlamydia when having contact to infected toilet seats. But for this to happen “fresh” infected secretes in high doses must be present. Chlamydia bacteria do not survive for long outside of a body.
Therefore an infection on this way is very unlikely and such a case hasn’t been reported.
For women an infection with chlamydia is especially dangerous since typical symptoms are often lacking. But leaving it untreated can pose significant hazards such as cancer and loss of fertility.
So, get yourself checked regularly.
Sources (German):
https://www.euroclinix.net/de/geschlechtskrankheiten/chlamydien/uebertragung
https://sicherintim.com/chlamydien-uebertragung/
I’d figure surface ones like herpes and crabs would be more likely, though maybe it depends on how sweaty you are and if there’s a moist path from that contact point to the opening. Probably practically impossible for male genitalia unless they are resting their junk on an infected surface, though even for female genitalia, I assume they don’t rest it right on the seat.