Does Rengar Legend ability use the stack?

Yes, the legend ability of Rengar, Pridestalker uses the chain.

According to the rules for triggered abilities:

• Triggered abilities (abilities starting with "When...", "Whenever...", or "At...") use the chain and can be reacted to.
• Since Rengar's ability reads "When you play a unit...", it is a triggered ability that is placed on the chain as a pending item whenever the condition (playing a unit) is met.

Because it uses the chain, it follows standard LIFO (Last-In, First-Out) resolution and players may respond to it with reactions.