I only have the Pikmin 3 Deluxe Demo, will that still count for speedruns?

Does the demo play through the whole game.

No, It has the tutorial area and The Garden Of Hope #1

The demo is allowed for runs of Tropical Forest on chr.mn/pik4 (the Mission Mode leaderboard for the Pikmin series). Other than that, I can't think of any categories you can use the demo for, besides Bad Ending, or maybe 6 fruits (but you wouldn't be able to do the fastest 6 Fruits route without access to Tropical Wilds).