I was up there yesterday. If you have a 4WD with high ground clearance you should be able to make it relatively far. I walked the road from the 4 mile mark (where most people park their 2WD cars) all the way out and past where you are likely considering driving. For your reference, it took about 55 minutes to walk it downhill after a long-ish day on the Crockers, Redington and Middle Redington Pond Peak.
The jeep road that branches off CVR and presumably goes up to the herd path is marked with a small cairn on the left and 2 pieces of orange flagging on trees leading you up the trail. I didn't follow this trail between CVR and the herd path near the summit, so I don't know for sure that this is the correct one, but I have read reports about it and it certainly looked like the right one when I was at both ends of it. (Take that for what it is worth.)
The metal grate bridge just after mile 4 has a couple wash-outs immediately after it that I didn't want to risk my car on.
Less than 1/2 a mile later there is an old bridge that people had to pile logs and 2x4 next to so their tires could get up onto it. I saw a jeep and a 4Runner had made it past this.
About 1 mile past the AT crossing there is a bridge marked impassable on the map where the road jogs west then back south. (Take the left fork!) This bridge may be passable but it looked pretty sketchy. People had cut some birch trees to fill in the gaps where the old logs had rotted out. It may take an ATV or a serious 4WD to get over this.
There is another bridge in the last mile or so that had a bad wash-out in front of the right ramp. Might be passable but all I know is at that point I wouldn't have considered driving that far up there in any vehicle that I ever owned.
There were other miscellaneous culverts, wash-outs, and small bridges along the rest of the way that didn't look so bad. The road gets kinda narrow back in there - lots of growth encroaching - but there were recent tire tracks, so someone had made it up there. The tracks were wider than an ATV but not big enough to be a logging truck.
My advice would be to just park at the 4 or 5.5 mile mark and walk the road. The time you will take to scope out the bridges and drive with care will not save you much over just grabbing your pack and walking it at 20 minute miles. Good luck!
