You know, I've always thought the same thing, that Vulcan was supposed to be a high-gravity planet. But I just checked the online transcript site, and that was never actually stated onscreen in TOS or the movies, or anywhere onscreen as far as I can tell. I checked the original proposal document from 1964 and the writers' bible from '67, and neither of them says anything about Vulcan's gravity. Nor does Spock's "biography" chapter in The Making of Star Trek
. The earliest reference that even comes close is in James Blish's adaptation of "Amok Time," where he refers to the tri-ox compound as a "high-G vitalizer shot" instead -- but even so, he has McCoy say it's only to compensate for the heat and thin air. Aside from that one word, he doesn't say anything about Vulcan's gravity.
The earliest reference I can find to Vulcan having high gravity is in The Star Fleet Medical Reference Manual
from 1977. There was also the 1980 USS Enterprise Officers Manual
by Geoffrey Mandel (one of the writers of the SFMRM) which said Vulcan's gravity was 1.251g
. (I haven't checked every source available, though; it would take too long to page through all the Bantam novels predating 1977 even if I still had them all.)
So there doesn't seem to be any official word on how Vulcan's gravity compares to Earth's. The high-gravity thing appears to be one of those bits of conventional wisdom that originated in unofficial sources or in fandom but have been taken for granted for so long that we don't realize they're not canonical. It's a logical (pardon the expression) conjecture that high strength would result from high gravity, but apparently that's all it is.
And "high gravity" is a relative term. I don't think anyone's ever proposed that Vulcan was a superterrestrial planet with gravity, say, 3 or 4 times that of Earth, anything really crushing. If Vulcan's gravity were 25% higher as Mandel proposed, then that would mean that a 160-pound man would only be carrying an extra 40 pounds of weight around -- equivalent to carrying a heavy backpack. Someone in good physical condition, as a Starfleet officer would be, could probably handle carrying such a load without showing obvious strain. And since that's a conjectural figure, it could be that Vulcan's gravity is more like 10% higher than Earth's, or some modest increase like that. Enough that it would tire people out more quickly but not enough to actually impair their mobility.