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How much time do we need in the Sydney airport, arriving from Los Angeles on Delta airlines, connecting to Virgin Australia to Queenstown, New Zealand? We have carry-on bags only.

Booked on one ticket, you formally require at least one hour.

Connecting at SYD
Incoming airline DL
Outgoing airline VA
Flight type International to International

STANDARD.D/D...D/I...I/D...I/I.
ONLINE .30 1.00 1.15 1.00
OFFLINE .30 1.00 1.15 1.00
** OR * ARE ALL
DL-VA II 1.00 TRM 1 - 1 
**-VA II 2.00 TRM 1 - 1

Across multiple tickets (or multiple PNRs), you will not be protected in the event your first flight is late, and you may need to re-arrange your transport at your own expense. (Formally, this is not a "connection" but two unrelated flights.) There is an international-international connection route at Sydney, so you won't need to go through customs or passports, but you will need to re-clear security.

Given that I assume you're on DL 41 from LAX, a fifteen hour flight over the Pacific, I would advise plenty of padding in your schedule. It only has an 80 % on time statistic, but the average delay is only twenty minutes. It's up to you what your risk appetite is; personally I would prefer at least two hours on one ticket (because even though the airline is obliged to rebook you at their expense, it is still disruptive and requires open seats) and perhaps three hours if I was taking the risk of buying a walk up flight to New Zealand. Ideally I would leave an overnight gap and plan to take a day in Sydney.