How can climbers gain access to Culebra Peak now that the mountain has new owners?

Yes.

Navigate to the small town of San Luis, Colorado. From here, you will turn east at the Conoco gas station. This is Rd P.6, travel on this for about 4 miles to the even smaller town of Chama, Colorado. At the intersection in Chama, turn left onto County Road L.7. Drive about 3.6 miles to County Road 25.5 which is right after a small bridge. Turn right and travel to the end of this small section of road (about .5 miles). From here, the gate is about a mile and has room for about 4-5 cars to park. The fee for hiking is $150 per person (2018).

Contact the new owners and tell them what you want to do and get their ideas.

I have done this many times before to get access to places that were private or under control of an assigned caretaker, ranger guy. I always just go "visit". Have a nice talk and explain how much you enjoy the outdoors and this place is special to you. Make it clear, (you have been here a long time in this section), you only want to visit and LEAVE nothing behind. You carry out all your trash. And would even take care of other things you may find he may not know of.

I also mention having another person around and out there can help find things that may need attention. Such as a garbage pile for instance. A gas well leaking or such. I can often weasel a hand written permission card stating I have authority to remove Trespassers as I see fit. WITH MUCH REASON of course, they only want to enjoy the property same as me. I only point out to others to be respectful and they are welcome if the owner says so. Go see the person. SIMPLE.

I suggest finding the owner and walking in for a talk with them and explain why this is so important to you that you were so bold.

give them money