You don't choose a target for Inti, Seneschal of the Sun's first ability at the time it triggers. Rather, a second "reflexive" ability triggers when you discard a card this way. You choose targets for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.

You pay all costs and follow all normal timing rules for cards played due to the last ability. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.