Are there any travel experiences in New Zealand that offer a Maori "cultural immersion"?

A tiny village on the central east coast of the north island is beginning to put together a native culture program, and while it is not really off the ground just yet, you could be one of the first to go and live with the Maori people of Porangahau and further this project by showing that there is interest. I will send your information to "Doc" whom I just met on a trip to Porangahau. I am working on a native culture project currently in Fiji, so we are trying to help each other further these programs. Look for answers here in the next few days, if Doc says it is OK, I will post his contact details here.

There is also the Tamaki experences Run by two of the Tamaki brothers. (Not their brother Bishop Brian Tamaki of Destiny Church) I went to the one in Christchurch at Ferrymead historic park it is very touristy apparently the one in Rotoura is less so. They also offer a (two/three) part interactive performance spread across their sites chronicling the family tree of a fictional chief.

Personally if you want to do it on the cheap head down to the grey river (South Island) or the Waikato river (North Island) during whitebait season. Find a pub near by and just chat with the old fellers when they take a break. For the price of a jug you'll probably get a good chat and maybe a free feed.

I am wondering if it's possible to travel to New Zealand as a tourist and immerse in Maori culture.

Yes. definitely. You may get something closer to what you see in the film "Whale Rider" than the 19th century images, but there are some "exceedingly hard core enclaves of Maoridom" that make substantial efforts to preserve the language and the culture. If you have the right attitude (as I'm sure you do have from what I've seen of yours so far) then you'll be able to find something which gets past the tourist level visits.

An area which so far probably tends not to feature on tourist level traipses is the Tuhoe area in the Ureweras. THese people have protested, promoted and defended their 'real Maoriness' in recent times with much collective national angst along the way, charges of terrorism (all dropped) armed offenders major activity (aka NZ version of SWAT teams) and more. And as a result we have probably all moved forwards quite a lot, not without some pain all round. I suspect that by the time you get here, looking up "Tuhoe" is liable to be a good start.

Tuhoe Maori Ureweras - images - you'll not be seeing any of the men in black balaclavas there that appear in some of these photos.

https://www.google.co.nz/sea

I think you'd be hard pressed to find a Marae with Maori living in the traditional way they would have before European settlers arrived, however Google provides plenty of results of Maraes offering accommodation up and down NZ.

You may or may not get the traditional cultural experience you are looking for, but no doubt you will meet some friendly characters and eat some good kai. Its still a side of NZ most tourists (and for a matter of fact most townies) won't have experienced.