The “westward penalty” from Kennedy/Canaveral would be about 800 m/s of delta-v, about 8-9% of the total delta-v requirement to orbit. Most crewed launchers intended for LEO to date have not had that much performance in reserve; a shortfall of only 100 m/s from LEO usually means prompt reentry.
Atlas/Mercury and Titan/Gemini could not have managed it. The boosters were completely expended to reach orbit, Gemini had only around 323 m/s of maneuvering delta-v, and that at a very low thrust-to-weight ratio, and Mercury had none.
The low Earth orbit Apollo missions (Apollo 7 and 9) could have reached orbit, and even carried out something like their intended missions.
Apollo 7 was a crewed CSM on a Saturn IB booster. The CSM had something like 2800 m/s available, with a fair thrust-to-weight ratio, and in fact the “mode IV” abort option would use the CSM as a third stage to reach orbit if the S-IVB second stage failed. Apollo 7 did a lot of orbital maneuvering to test the CSM engine, and that would have had to be cut short if it were going to spend that much fuel on ascent, but a sizable portion of the original mission plan could have been carried out in retrograde.
Apollo 9 was a CSM/LM flight to LEO on a Saturn V; if fully fueled, the third stage would have had around 3000 m/s of delta-v capability (needed for translunar flight), so even a much lighter fuel load would suffice to go into retrograde orbit.
I believe any of the Apollo lunar missions could have gone from a retrograde Earth orbit ascent to a lunar flyby without hardware modification, abandoning the LM (or docking and extracting it very quickly) when the S-IVB ran out of fuel and completing the TLI on the CSM’s engine. The delta-v budget to enter lunar orbit and then return to Earth is around 1400 m/s. If the LM wasn’t brought along (as on Apollo 8) and both the S-IVB and CSM were fully fueled, a lunar orbit mission might even have been possible from retrograde LEO.
The space shuttle should have been able to do it if a light payload was carried. My quick-and-dirty spreadsheet estimation says it would carry something like 4 tons to retrograde LEO instead of the 27 tons possible into prograde LEO. Unlike the Saturns, the shuttle did have a West coast launch site, never used, that would have made a retrograde launch practical if it had been needed.
I don’t think the Soyuz or Long March launchers had that sort of performance margin.