Depending on the snow pack most of the higher trails may still be closed in early June. There's no "wheeling" in either Yellowstone or RMNP anyway, although there are a few dirt roads (Fall River Road in RMNP is a nice drive but may not be open yet in June.)
Remember that most of the high country 4wd roads don't open until late June or even early July. In some years there are some roads that never open. I've been blocked by a snowdrift on Mosquito Pass (13,180', the highest continuous road in North America) in mid August.
If you want to see wildlife at Yellowstone your best bet is to get there early in the day. When we went in July of 2013, we camped in Cody (about 50 miles East of the East entrance of Yellowstone) and woke up at 4am to drive in. We were able to see wolves and bears and there weren't many crowds because most people had not gotten up yet.
If you're staying in West Yellowstone or Gardiner (just North of the park in Montana) you won't have to get up as early but I'd try to get to the prime viewing sites before sunrise if that's what you want to see.
Also you said "the 6 days we will actually be there" do you mean you will have a total of 6 days at both parks? Hopefully you accounted for the fact that driving from RMNP to Yellowstone or vice-versa will eat up an entire day or more, depending on where you stay. It's at least a 500 mile trip or more.