5e only offers height ranges for two categories, but from that, we can extrapolate the remaining ranges
After some (detailed) digging, I found that the Players Handbook lists two sets of measurements for the size categories that Player Characters fit into:
Characters of most races are Medium, a size category including creatures that are roughly 4 to 8 feet tall. Members of a few races are Small (between 2 and 4 feet tall), which means that certain rules of the game affect them differently. [...]
PHB > Chapter 2: Races > Racial Traits > Size
Helpfully, the wording of this tells us describes the general size category ranges, as opposed to just the size category as it applies to Player Characters. So this gives us a starting point for our Height calculation.
We can then use the text of the Enlarge/Reduce spell:
Enlarge. The target's size doubles in all dimensions, and its weight is multiplied by eight. This growth increases its size by one category - from Medium to Large, for example.
Reduce. The target's size is halved in all dimensions, and its weight is reduced to one-eighth of normal. This reduction decreases its size by one category - from Medium to Small, for example.
So if we double our upper and lower bounds we should get the Height ranges for size categories above Medium, and if we halve our upper and lower bounds we should get the height ranges for categories below Small.
First we should sanity check this method, using our two known size categories, and thankfully it works as expected with those upper and lower bounds!
Extrapolating this, and using the size categories we have defined in the rules for 5e (Tiny, Small, Medium, Large, Huge, and Gargantuan) we get the following table:
Technically, given the fact Tiny and Gargantuan are the minimum and maximum size categories respectively, their respective lower and upper bounds wouldn't have limits on them as shown in the above table. I found it useful, however, to include them, as this method would also enable us to generate additional size categories above and below the default given ones in the 5e rules.
Weight, is trickier, as the canonical Height and Weight tables, showing in the PHB, Volo's Guide to Monsters, and Guildmaster's Guide to Ravnica (as well as other 1-2 new race specific source books), list out weights of playable races. We can use these as guideposts, but the ranges are a lot fuzzier and require a lot more work. I'll update this answer when I've collated that information.