Can I play a reaction when an enemy plays a Carnivorous Snapvine to their base targeting a unit at a battlefield I control?

Yes, you can react to the on-play ability of Carnivorous Snapvine (and similar units like Riptide Rex) with Reaction-speed spells. Here's how it works: When the unit is played, it enters the board first. Then the Play Effect goes on the chain as a Pending Item. The controlling player completes the steps of playing it, including targeting. Once it becomes a Chain Item, it functions just like a spell. Other players have an opportunity to play Reactions before the resolution of spells. Since Play Effects become Chain Items 'just like a Spell,' they also have a reaction window before they resolve. So you cannot react to the unit being played itself, but you CAN react to the on-play ability before it resolves, using Reaction-speed spells only.