I'd imagine the US could whip up their own Lunakhod (a very successful machine incidently) by the later 60s, but what would be the point? They'd already done hard and soft robotic landings and were on the verge of manned landings.
I guess that what you'd need would be for the US to abandon the manned program and shift to robotic landings to continue exploring the moon.
So, POD:
Apollo XII without a Steely-eyed Missile Man.
36 seconds after lift-off, Apollo 12 was struck by lightning. The power surge caused instruments to malfunction and telemetry to become nonsensical. The flight director thought that he would have to abort the mission, but in OTL, John Aaron realised the he had seen something like this before and could tell the crew which switch to flick in order to return the instruments to correct functioning, thus saving the mission. For this, he received NASA's highest non-official accolade of being called a "Steely-Eyed Missile Man".
So WI Aaron doesn't remember this fix. The Apollo 12 mission is abandoned. By this time, Apollo 13 was already under construction, complete with faults, so was likely to have the same explosion as in OTL. However butterflies mean that conditions are slightly worse than in OTL and the crew die.
So, of three missions, we have one success and two failures (one fatal). There will be plenty of people (including some Senetors) who will say that Apollo has achieved it's aim of landing a man on the moon and there is no need to continue such risky operations. At this time, NASA doesn't have much in the way of plans for what to do after Apollo, and those plans it does have assumed that Apollo would be followed by further manned missions. Instead, NASA will have to switch to more non-manned space exploration, including more robotic landings on the moon.
Cheers,
Nigel.