Assuming he runs in 1968, he can probabily win a fairly good plurality. However, he nonetheless sees domestic trouble in his second (1965-1969) term due to racial tensions and the issue of open housing. This makes '68 much narrower (already narrower by merit of not being 1964), though I think Johnson can pull it off in that environment. Who knows how he handles Nixon's OTL first term though. I'd guess that, facing recession and Democratic fatigue, as well as being aided by a new Southern Strategy, the GOP is back in '72.
I think he could've won in '68 even with Vietnam. Look at how close it was with Humphrey. Johnson knew how to win and may have been able to bring peace in the fall of 1968. It would've been a peace similar to what we ended up getting in 1973, but at least would've been 5 years earlier.
But he decided pretty early that he didn't want to run in 1968, long before Vietnam became an issue--possibly as early as 1965 and definitely by 1967. He truly believed he would die by age 64--which he did. He actually had to be pushed by his family and those close to him to run in 1964 as it was. He had a whole speech denying his nomination in 1964 written out.