I would go in person. A letter out of the blue might work, but if I got something like that in the mail, I'd be creeped out...I agree with Hinmo, probably the best way to approach them is to look for them when they're outside. This is how I sometimes used to ask permission to hunt deer on someone else's property. Along those lines, do you know someone who knows the landowners? That can be a great way to get in touch with them, since you have someone known to both of you involved. This is going to take some time. Be patient. Be prepared for when you make contactwith the landowner. Bring Photos of your bonsai along with a coherent explanation of why you want to dig up trees on their property. You're going to have to explain bonsai to them a bit. The mention of "bonsai" on their property CAN lead to assumptions by some landowners that they are somehow sitting on some kind of gold mine with valuable bonsai trees. Some landowners will be cool about it, others not. Be prepared for rejection and quizzical looks. You might offer some cash in exchange for the trees--like $25 or something, if you get resistance. Above all be humble and polite. Assure them you will not harm anything on their property--this include FILLING IN HOLES completely. Holes in a used pasture can break cows' legs. Close gates behind you.
and FWIW, at first skip asking to air layer stuff on someone else's property. You're going to have to regularly go on their land to take care of those, which can be a big obstacle. People aren't really going to understand WTF air layering is. That might be down the road with landowners who let you collect trees...Pick your battles...