Once an ability that causes a creature to connive begins to resolve, no player may take any other actions until it's done. Notably, opponents can't try to remove the conniving creature after you discard cards but before it receives +1/+1 counters, if any.

If no cards are discarded, most likely because that player's hand is empty and an effect says they can't draw cards, the conniving creature does not receive any +1/+1 counters.

Connive X is a variant of connive. If a creature connives X, its controller will draw X cards, discard X cards, then then put a number of +1/+1 counters on the conniving permanent equal to the number of nonland cards discarded this way.

If a resolving spell or ability instructs a specific creature to connive but that creature has left the battlefield, the creature still connives, although you can't put any +1/+1 counters on it. Abilities that trigger "when [that creature] connives" will trigger.

You must already control a Swamp as Spymaster's Vault enters the battlefield for it to enter untapped. If it enters the battlefield at the same time as a Swamp when you control no other Swamps, it will enter tapped.