Can I play official matchmaking if I don't buy Legacy of the Void?

First of all: LOTV is a stand-alone game, you dont need the previous 2 games to play it. That might resolve your question alreay. If not, read further.

Every expansion has it's own matchmaking, so if you only have HOTS, you will get matched vs other HOTS players. The same goes for WOL.

There are still people playing HOTS and LOTV as stated in this reddit thread, so if you only want to get one of those, you should be fine.
But this thread is 6 months old, so things might have shifted a bit. WOL has probably lost a lot more players with the release of LOTV. You should be fine for lower levels as far as queue times go, but most of the higher skilled player have probabably moved to LOTV, so I expect queue times to be quite long after platinum.
HOTS however should be fine, since LOTV is only out for less than 6 months.

If you choose to get only WOL ot HOTS, you should also keep the following points in mind:

• No new maps.
• No balance patches (but at least swarmhosts are already fixed).
• No new content.
• Very stale meta, because of the points above.
• All tournaments are played in LOTV, so there won't be any pros/ streamers that play WOL or HOTS.

My recommendation:
Get the starter edition first. Starcraft is a hard game and not for everyone. If you tried it out and like it, I would strongly recommend getting LOTV. As mentioned, you dont need to buy WOL and HOTS for that. I wouldn't really recommend getting only WOL and HOTS, since both together are the same cost as LOTV.