Best place on Fraser Island in 2020 to use a caravan as a base?

Plenty of vans go up the beach. Any of the beach zones south of Indian head would be advised for the van.
If the tyres don’t track behind your car I’d suggest going down very low in the pressure department for the inskip, barge, hook point section then add a bit for the beach run. Especially if the van is on trailer sized tyres rather than car/4WD sized tyres. It is often the typical prado towing the Jayco swan style setup that gets caught out at inskip due to mainly trailer track width and not compensating for it with pressure. I towed boat trailers with similar dimensions up there for a few years. I now have LT265 75 r 16 on both car and trailer (2 tonne and same wheel track) and run 18 psi as a general guide. If your van is on 185 r 14s or similar you may not be able to run that low for a long period of time hence the above comment.
March flies are fierce up there at that time, as are tourists haha. You’ll have a ball. Tyre pressure beats right boot every time.

We camped at Dundubara back in the mid 1990’s. It is just off the beach south of Indian Head, and has good sites back then. Not sure what it is like now.

Central Station is also a good camping ground, but because it is very shady, it is no good if you rely on solar for battery recharging.

Macca.

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Fraser Island is a National Park controlled by the QPWS&P. If you have not already seen it, here is their webpage.
You are only permitted to camp in the designated sites or in the several commercial sites and bookings are required.
Of course, bookings are heavy for the New Year period and it is likely that there are now no sites available along the Eastern beach. You would find it very difficult (impossible?) to reach the unserviced sites on the Western beach and in any case, they are also probably booked out for NY.

Apart from availability, with your rig I would opt to stay at the commercial Cathedrals on Fraser Which is on the Eastern Beach, a bit more than halfway up the island.

Good luck.

Spent 12 days there 2 weeks ago camped on the Eastern beach in a 17ft dual axle Jayco Expanda. My advice is go via Inskip only, don’t try and drag a van across the island, it’s rough and slow going and skinny tracks with oncoming vehicle will cause issues...pick a zone somewhere North of Eurong and up to about Dundaburra to be central to most “attractions” on the island. You could go abit North of Dundaburra also, there was heaps of vacant sites there in that zone over the school holidays
Time run up the island for 2 hours before low tide, this usually gives plenty of hard sand to drive on and allows you to drive around Poyungan and Yidney rocks rather then drag a van through their bypass tracks, avoids the big holes, rocks to go over and steep step ups you can find on these bypasses.
Inskip I drop vehicles rear tyres to 15psi and fronts to 12psi, caravan tyres go 14psi alround...rather go lower then spend an hour trying to unbog my set up, has worked the past few years for me with no bogging. Have towed various caravans there over the years without issue following that guideline
As an idea...I also take 2 empty jerry cans and 2 x 20 ltr buckets with lids on them that I collect water with for the caravan for showers etc, a mate takes an empty 80ltr water tank and transfer pump that he uses to refill his vans water supply...there’s various free places to get water (Dundaburra campground is one place for example) and some places like Cathedral from memory charge $2 per 20ltrs

Only been once, but Dilli village was a great spot to camp. Sure, it's further to the "attractions", but running at low tide ( when else would you go ) still left plenty of time to get everywhere .