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 an extra 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.
Of course, any of the launchers for these missions could simply have been scaled up slightly to provide the additional needed performance.