I have a quick question about unlocking access to a race/variant/sub-race that's available in multiple books, specifically; can I get a race by purchasing any book that contains it, or can I only get a race from the book it is shown under on the races page?

For example; the Kenku race is listed on that page only under Volo's Guide to Monsters, but I know that full details for this race can also be found in the Explorer's Guide to Wildemount, so I'm wondering; if I purchase Explorer's Guide to Wildemount, will I gain access to the Kenku race, or must I purchase it from Volo's Guide to Monsters (either individually, with all races or as the entire book)?

Likewise that same book includes the Tortle race, but not any of its variants; by purchasing that book or its races, would I gain access to regular Tortles, or must that be purchased only through the Tortle package?

I don't own Volo but I do own wildemont. According to the purchase page I own kenku so yeah, looks like it works if you own wildemount. 

If you purchase a book, you get access to any PC races listed in that book.  Though that can cause some variations- orcs, for example have slightly different stats depending on whether you're using Volo's Guide, Eberron, or Wildemount and the character creator lists all three separately (Orc, Orc of Eberron, and Orc of Wildemount, respectively).

To buy piecemeal, e.g. just the race, it usually is only available from the first book where it is featured.

Typically if you buy a whole sourcebook that contains the options such as races published in previous books, however, you still get those options.